This dissertation is concerned with the study of global inverse mapping theorems and surjectivity theorems in smooth and nonsmooth analysis.

First, we consider the problem of global solvability of $\alpha$-expanding maps in reflexive Banach spaces and provide, utilizing a recent generalization of the interior mapping theorem, a positive answer to an open problem stated by Nirenberg. After that, we introduce the concept of a ray-proper map and prove, using a method given by F. John, a generalization of the Banach-Mazur theorem. We also prove some global invertibility theorems for locally Lipschitz-continuous maps defined on Euclidean finite dimensional spaces; in particular, we establish a generalization of the Hadamard theorem for locally Lipschitz-continuous maps. Finally, we prove some global and local inverse function theorems for certain classes of set-valued maps; as a by-product, we obtain a new surjectivity theorem for single-valued maps.