Generalized Weyl Algebras (GWAs) are ring extensions that encompass a wide range of noncommutative algebras, including the Weyl algebra, quantum groups, and certain Lie algebras. In this talk I will discuss their representation theory, with emphasis on a new class of R-free modules obtained by lifting modules that are free over the base ring. I will discuss the classification of such modules, and in the case where R is a PID, present a combinatorial description of their composition series. I will also show how these R-free modules are connected to the classical weight modules over GWAs. The talk is based on joint work with Samuel Lopes.