Abstract: Characteristic classes of manifolds are usually cohomology classes associated to the tangent bundle of the manifold. Hirzebruch's theory unifies three important theories of characteristic classes of smooth manifolds: Chern classes, Todd classes and Thom-Hirzebruch L-classes. In 2010, Brasselet, Schürmann and Yokura extended Hirzebruch classes to singular varieties using natural transformations, and unified in a functorial sense the three theories of Chern classes, Todd classes and L-classes for singular varieties. In this talk, we will delve into the relationships between these classes.