Manifolds with Cusps of Rank One : Spectral Theory and L2 Index Theorem The manifolds investigated in this monograph are generalizations of (XX) rank one locally symmetric spaces. In the first part of the book the author develops spectral theory for the differential Laplacian operator associated to the so called generalized Dirac operators on manifolds with cusps of rank one. This includes the case of spinor Laplacians on (XX) rank one locally symmetric spaces. The time dependent approach to scattering theory is taken to derive the main results about the spectral resolution of these operators. The second part of the book deals with the derivation of an index formula for generalized Dirac operators on manifolds with cusps of rank one. This index formula is used to prove a conjecture of Hirzebruch concerning the relation of signature defects of cusps of Hilbert modular varieties and special values of L series. This book is intended for readers working in the field of automorphic forms and analysis on non compact Riemannian manifolds and assumes a knowledge of PDE scattering theory and harmonic analysis on semisimple Lie groups