  In this paper, by introducing the degenerate Fubini-type polynomials, we give
several relations with the help of the Fa\`a di Bruno formula and some
properties of Bell polynomials, and generating function methods. Also, we
derive some new explicit formulas and recurrence relations for Fubini-type
polynomials and numbers. Associating the degenerate Fubini-type polynomials
newly defined here with degenerate Apostol-Bernoulli polynomials and degenerate
Apostol-Euler polynomials of order {\alpha} enables us to present additional
relations for some degenerate special polynomials and numbers.
