<PAGE>   1

    As filed with the Securities and Exchange Commission on August 31, 1998
                                                       Registration No. 33-72602
================================================================================
                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549
                                  ------------
                         POST-EFFECTIVE AMENDMENT NO. 1
                                       TO
                                    FORM S-8

                             REGISTRATION STATEMENT
                                      UNDER
                           THE SECURITIES ACT OF 1933
                                  ------------
                                  AVIALL, INC.
             (Exact name of registrant as specified in its charter)

                               2075 DIPLOMAT DRIVE
                            DALLAS, TEXAS 75234-8999
                                 (972) 406-2000
               (Address, including zip code, and telephone number,
        including area code, of registrant's principal executive offices)

        DELAWARE                                   65-0433083
 (State of incorporation)            (I.R.S. Employer Identification Number)

                     AVIALL, INC. 1998 DIRECTORS STOCK PLAN
                            (Full title of the plan)

                             JEFFREY J. MURPHY, ESQ.
                 SENIOR VICE PRESIDENT, LAW AND HUMAN RESOURCES,
                          SECRETARY AND GENERAL COUNSEL
                                  AVIALL, INC.
                               2075 DIPLOMAT DRIVE
                            DALLAS, TEXAS 75234-8999
                                 (972) 406-2000
            (Name, address, including zip code, and telephone number,
                   including area code, of agent for service)

                                 WITH A COPY TO:

                             JAMES E. O'BANNON, ESQ.
                           JONES, DAY, REAVIS & POGUE
                            2300 TRAMMELL CROW CENTER
                                2001 ROSS AVENUE
                               DALLAS, TEXAS 75201
                                 (214) 220-3939
                                   -----------
                         CALCULATION OF REGISTRATION FEE

<TABLE>
<CAPTION>
===================================================================================================================================
                                                                                Proposed           Proposed
                                                                                 Maximum            Maximum
Title of                                                        Amount          Offering           Aggregate           Amount of
Securities to                                                   to be           Price per          Offering          Registration
be Registered                                               Registered(2)       Share(3)            Price(3)            Fee(3)
-----------------------------------------------------------------------------------------------------------------------------------
<S>                                                         <C>                 <C>                <C>               <C>          
Common Stock, par value $0.01 per share(1).  .  .  .        135,000 shares      $ 13.15625         $ 1,776,094       $ 524
===================================================================================================================================

</TABLE>

  (1) This Registration Statement also relates to rights to purchase Series A
      Junior Participating Preferred Stock, par value $.01 per share, of Aviall,
      Inc. ("Rights"). One Right will be issued together with and will attach to
      each share of Common Stock, par value $.01 per share, of Aviall, Inc.
      issued pursuant to the Aviall, Inc. 1998 Directors Stock Plan.
  (2) Plus such indeterminate number of shares as may be issued to prevent
      dilution resulting from stock splits, stock dividends or similar
      transactions in accordance with Rule 416 under the Securities Act of 1933,
      as amended.
  (3) Computed in accordance with Rule 457(h)(1).
================================================================================



<PAGE>   2







                                     PART I

              INFORMATION REQUIRED IN THE SECTION 10(a) PROSPECTUS


                                EXPLANATORY NOTE

         The information called for by Part I of this Post-Effective Amendment
No. 1 to Registration Statement on Form S-8 (the "Registration Statement") is
included in the description of the Aviall, Inc. 1998 Directors Stock Plan (the
"Plan") to be delivered to persons eligible to participate in the Plan. Pursuant
to the Note to Part I of Form S-8, this information is not being filed with or
included in this Registration Statement.

                                      I-1

<PAGE>   3




                                     PART II

               INFORMATION REQUIRED IN THE REGISTRATION STATEMENT

ITEM 3.  INCORPORATION OF DOCUMENTS BY REFERENCE.

         The following documents (or, as applicable, the portions thereof
specified below) filed by Aviall, Inc. (the "Company") with the Securities and
Exchange Commission (the "Commission") pursuant to the Securities Exchange Act
of 1934, as amended (the "Exchange Act"), are incorporated by reference, as of
their respective dates, in this Registration Statement: (i) the Company's Annual
Report on Form 10-K for the year ended December 31, 1997 (Commission File No.
1-12380); (ii) the Company's Quarterly Report on Form 10-Q for the quarter ended
March 31, 1998; (iii) the Company's Quarterly Report on Form 10-Q for the
quarter ended June 30, 1998; and (iv) the description of the Common Stock
contained in the Company's Registration Statement on Form 10, as amended, filed
with the Commission pursuant to Section 12(b) of the Exchange Act on December
22, 1993.

         In addition, all documents hereafter filed by the Company pursuant to
Sections 13(a), 13(c), 14 or 15(d) of the Exchange Act prior to the filing of a
post-effective amendment which indicates that all securities offered hereby have
been sold or which deregisters all such securities then remaining unsold shall
be deemed to be incorporated by reference in this Registration Statement and to
be a part hereof from the date of filing of such documents.

         Any statement contained herein or in a document incorporated or deemed
to be incorporated by reference herein shall be deemed to be modified or
superseded for purposes of this Registration Statement to the extent that a
statement contained herein or in any subsequently filed document which also is
or is deemed to be incorporated by reference herein modifies or supersedes such
statement. Any statement so modified or superseded shall not be deemed, except
as so modified or superseded, to constitute a part of this Registration
Statement.

ITEM 4.  DESCRIPTION OF SECURITIES.

         Not applicable (securities to be offered are registered under Section
12 of the Exchange Act).

ITEM 5.  INTERESTS OF NAMED EXPERTS AND COUNSEL.

         Not Applicable.

ITEM 6.  INDEMNIFICATION OF DIRECTORS AND OFFICERS.

         Section 145 of the Delaware General Corporation Law ("Delaware Law")
permits indemnification of the directors and officers of the Company involved in
a civil or criminal action, suit or proceeding, including, under certain
circumstances, suits by or in the right of the Company, for any expenses,
including attorney's fees, and (except in the case of suits by or in the right
of the Company), any liabilities which they may have incurred in consequences of
such action, suit or proceeding under conditions stated in said Section.

         Article XI ("Article XI") of the Company's Restated Certificate of
Incorporation (the "Certificate") limits the personal liability of the Company's
directors to the Company or its stockholders for monetary damages for breach of
fiduciary duty. In addition, Article VI ("Article VI") of the Company's By-Laws
(the "By-Laws") defines and clarifies the rights of certain individuals,
including the Company's directors and officers, to indemnification by the
Company against personal liability or expenses incurred by them as a result of
certain litigation against them.

         Set forth below is a description of Article XI and Article VI. Such
descriptions are intended as summaries only and are qualified in their entirety
by reference to the Company's Certificate and the Company's By-Laws, copies of
which have been filed as exhibits to this Registration Statement.

                                      II-1

<PAGE>   4



         Article XI protects the directors against personal liability for
breaches of the duty of care. Such Article absolves directors of liability for
negligence in the performance of their duties, including gross negligence.
Directors remain liable for breaches of the duty of loyalty to the Company and
its stockholders as well as for acts or omissions not in good faith or which
involve intentional misconduct or a knowing violation of law and transactions
from which a director derived improper personal benefit. In addition, Article XI
does not absolve directors of liability for unlawful dividends or stock
repurchases or redemptions to which a negligence standard presently applies
under Delaware Law. Also, there may be certain liabilities, such as those under
the federal securities laws or other state or federal laws, which a court may
hold are unaffected by Article XI.

         Although Article XI provides the directors with protections against
personal liability for monetary damages for breaches of the duty of care, it
does not eliminate the directors' duty of care. Accordingly, Article XI would
have no effect on the availability of equitable remedies such as an injunction
to prevent a proposed action or rescission of a contract based upon a director's
breach of the duty of care. Although both directors and officers of the Company
are covered by indemnification provisions under Article VI (see below), Article
XI limits liability only with respect to a person acting in the capacity of a
director.

         Article VI provides that each person who was or is made a party to, or
is involved in any action, suit or proceeding by reason of the fact that he or
she is or was a director, officer or employee of the Company (or was serving at
the request of the Company as a director, officer or employee of another entity,
including service with respect to employee benefit plans maintained or sponsored
by the Company) will be indemnified and held harmless by the Company, to the
fullest extent authorized by Delaware Law, as currently in effect (or, to the
extent indemnification in broadened, as it may be amended) against all expense,
liability or loss (including attorneys' fees, judgments, fines, ERISA excise
taxes or penalties and amounts to be paid in settlement) reasonably incurred by
such person in connection therewith. Article VI provides that the rights
conferred therein are contract rights and include the right to be paid by the
Company for the expenses incurred in defending the proceedings specified above,
in advance of their final disposition, except that, if Delaware Law so requires,
such payment will only be made upon delivery to the Company by the indemnified
party of an undertaking to repay all amounts so advanced if it is ultimately
determined that the person receiving such payments is not entitled to be
indemnified under the By-Laws or otherwise. Article VI provides that the Company
may, by action of its Board of Directors, provide indemnification to its agents
with the same scope and effect as the foregoing indemnification of directors,
officers and employees.

         Article VI provides that persons indemnified thereunder may bring suit
against the Company to recover unpaid amounts claimed thereunder, and that if
such suit is successful, the expense of bringing such a suit will be reimbursed
by the Company. Article VI further provides that while it is a defense to such a
suit that the person claiming indemnification has not met the applicable
standards of conduct making indemnification permissible under Delaware Law, the
burden of proving the defense will be on the Company and neither the failure of
the Company's Board to have made a determination that indemnification is proper,
nor an actual determination by the Company's Board that the claimant has not met
the applicable standard of conduct, will be a defense to the action or create a
presumption that the claimant has not met the applicable standard of conduct.

         Article VI provides that the rights to indemnification and the payment
of expenses incurred in defending a proceeding in advance of its final
disposition conferred therein will not be exclusive of any other right which any
person may have or acquire under any statute, provision of the Company's
Certificate of Incorporation or the Company's By-Laws, or otherwise. Article VI
also provides that the Company may maintain insurance, at its expense, to
protect itself and any of its directors, officers, employees or agents against
any expense, liability or loss, whether or not the Company would have the power
to indemnify such person against such expense, liability or loss under Delaware
Law.

ITEM 7.  EXEMPTION FROM REGISTRATION CLAIMED.

         Not Applicable.


                                      II-2

<PAGE>   5



ITEM 8.  EXHIBITS.

         4.1      --   Restated Certificate of Incorporation of the Company
                       (incorporated by reference to Exhibit 3.1 to the
                       Company's Annual Report on Form 10-K for the fiscal year
                       ended December 31, 1993 (the "1993 Form 10-K"))

         4.2      --   By-Laws of the Company (incorporated by reference to
                       Exhibit 3.2 to the 1993 Form 10-K)

         4.3      --   Amendment to the By-Laws of the Company

         4.4      --   Aviall, Inc. Preferred Stock Purchase Rights Plan
                       between the Company and The First National Bank of Boston
                       dated as of December 7, 1993 (incorporated by reference
                       to Exhibit 10.7 to the 1993 Form 10-K)

         5.1      --   Opinion of Jones, Day, Reavis & Pogue

         23.1     --   Consent of PricewaterhouseCoopers LLP

         23.2     --   Consent of Jones, Day, Reavis & Pogue (included in
                       Exhibit 5.1)

         24.1     --   Power of Attorney

         99.1     --   Aviall, Inc. 1998 Directors Stock Plan (incorporated
                       by reference to Exhibit 10.1 to the Company's quarterly
                       report on Form 10-Q for the quarter ended June 30, 1998)

ITEM 9.  UNDERTAKINGS.

         A.  The Company hereby undertakes:

                   (1) To file, during any period in which offers or sales are
                  being made, a post-effective amendment to this Registration
                  Statement: (i) to include any prospectus required by Section
                  10(a)(3) of the Securities Act of 1933, as amended (the
                  "Securities Act"), unless the information required to be
                  included in such post-effective amendment is contained in
                  periodic reports filed by the Company pursuant to Section 13
                  or Section 15(d) of the Exchange Act and incorporated herein
                  by reference; (ii) to reflect in the prospectus any facts or
                  events arising after the effective date of this Registration
                  Statement (or the most recent post-effective amendment
                  thereof) which, individually or in the aggregate, represent a
                  fundamental change in the information set forth in this
                  Registration Statement, unless the information required to be
                  included in such post-effective amendment is contained in
                  periodic reports filed by the Company pursuant to Section 13
                  or Section 15(d) of the Exchange Act and incorporated herein
                  by reference; and (iii) to include any material information
                  with respect to the plan of distribution not previously
                  disclosed in this Registration Statement or any material
                  change to such information in this Registration Statement;
                  provided, however, that paragraphs (A)(1)(i) and (A)(1)(ii) do
                  not apply if the information required to be included in a
                  post-effective amendment by those paragraphs is contained in
                  periodic reports filed with or furnished to the Commission by
                  the Company pursuant to Section 13 or 15(d) of the Exchange
                  Act that are incorporated by reference in this Registration
                  Statement;

                   (2) That, for the purpose of determining any liability under
                  the Securities Act, each such post-effective amendment shall
                  be deemed to be a new registration statement relating to the
                  securities offered therein, and the offering of such
                  securities at that time shall be deemed to be the initial bona
                  fide offering thereof; and

                   (3) To remove from registration by means of a post-effective
                  amendment any of the securities being registered which remain
                  unsold at the termination of the offering.

                                      II-3
<PAGE>   6



         B. The Company hereby undertakes that, for purposes of determining any
liability under the Securities Act, each filing of the Company's annual report
pursuant to Section 13(a) or Section 15(d) of the Exchange Act that is
incorporated by reference in this Registration Statement shall be deemed to be a
new registration statement relating to the securities offered therein, and the
offering of such securities at that time shall be deemed to be the initial bona
fide offering thereof.

         C. Insofar as indemnification for liabilities arising under the
Securities Act may be permitted to directors, officers and controlling persons
of the Company pursuant to the foregoing provisions, or otherwise, the Company
has been advised that in the opinion of the Commission such indemnification is
against public policy as expressed in the Securities Act and is, therefore,
unenforceable. In the event that a claim for indemnification against such
liabilities (other than the payment by the Company of expenses incurred or paid
by a director, officer, or controlling person of the Company in the successful
defense of any action, suit or proceeding) is asserted by such director,
officer, or controlling person in connection with the securities being
registered, the Company will, unless in the opinion of its counsel the matter
has been settled by controlling precedent, submit to a court of appropriate
jurisdiction the question whether such indemnification by it is against public
policy as expressed in the Securities Act and will be governed by the final
adjudication of such issue.


                                      II-4

<PAGE>   7

                                 SIGNATURES

         Pursuant to the requirements of the Securities Act of 1933, the
Registrant certifies that it has reasonable grounds to believe that it meets all
of the requirements for filing on Form S-8 and has duly caused this
Post-Effective Amendment No. 1 to Registration Statement on Form S-8 to be
signed on its behalf by the undersigned, thereunto duly authorized, in the City
of Dallas, State of Texas on this 31st day of August, 1998.

                                   AVIALL, INC.

                                   By: /s/ JEFFREY J. MURPHY
                                      -----------------------------------
                                           Jeffrey J. Murphy
                                       Senior Vice President, Law and Human
                                     Resources, Secretary and General Counsel

        Pursuant to the requirements of the Securities Act of 1933, this
Post-Effective Amendment No. 1 to Registration Statement on Form S-8 has been
signed below by the following persons in the capacities indicated on this 31st
day of August, 1998.

             SIGNATURE                                     TITLE

      /s/ ERIC E. ANDERSON                     Chairman, President and Chief 
------------------------------------                 Executive Officer
          Eric E. Anderson                      (Principal Executive Officer)
         
                                                                         
      /s/ JACQUELINE K. COLLIER                Vice President and Controller)
------------------------------------            (Principal Accounting Officer)
          Jacqueline K. Collier

      /s/ CORNELIUS VAN DEN HANDEL             Vice President and Treasurer
------------------------------------            (Principal Financial Officer)
          Cornelius Van Den Handel              


          ROBERT G. LAMBERT*                            Director
------------------------------------
          Robert G. Lambert                


          HENRY A. MCKINNELL*                           Director
------------------------------------
          Henry A. McKinnell


          DONALD R. MUZYKA*                             Director
------------------------------------
          Donald R. Muzyka


          RICHARD J. SCHNIEDERS*                        Director
------------------------------------
          Richard J. Schnieders
                                                                       

          BRUCE N. WHITMAN*                             Director
------------------------------------
          Bruce N. Whitman
                                                                      

*     The undersigned, by signing his name hereto, does sign and execute this
      Post-Effective Amendment No. 1 to Registration Statement on Form S-8
      pursuant to Power of Attorney executed by the above-named directors and
      officers of the Registrant and contemporaneously filed herewith with the
      Securities and Exchange Commission.



                                       /s/ JEFFREY J. MURPHY
                                      ----------------------------------------
                                           Jeffrey J. Murphy,
                                           Attorney-in-Fact



<PAGE>   8







                                INDEX TO EXHIBITS

<TABLE>
<CAPTION>
Exhibit 
No.              Exhibit
-------          -------
 <S>             <C>
 4.1     --      Restated Certificate of Incorporation of the Company
                 (incorporated by reference to Exhibit 3.1 to the Company's
                 Annual Report on Form 10-K for the fiscal year ended December
                 31, 1993 (the "1993 Form 10-K"))

 4.2     --      By-Laws of the Company (incorporated by reference to Exhibit
                 3.2 to the 1993 Form 10-K)

 4.3     --      Amendment to the By-Laws of the Company

 4.4     --      Aviall, Inc. Preferred Stock Purchase Rights Plan between
                 the Company and The First National Bank of Boston dated as of
                 December 7, 1993 (incorporated by reference to Exhibit 10.7 to
                 the 1993 Form 10-K)

 5.1     --      Opinion of Jones, Day, Reavis & Pogue

 23.1    --      Consent of PricewaterhouseCoopers LLP

 23.2    --      Consent of Jones, Day, Reavis & Pogue (included in Exhibit 5.1)

 24.1    --      Power of Attorney

 99.1    --      Aviall, Inc. 1998 Directors Stock Plan (incorporated by
                 reference to Exhibit 10.1 to the Company's quarterly report on
                 Form 10-Q for the quarter ended June 30, 1998)
</TABLE>