This book provides an updated discussion of fixed point theory using the framework of p-vector spaces, a core component of nonlinear analysis in mathematics. The book covers three main topics: 1) the 'best approximation approach' for classes of semiclosed 1-set contractive set-valued mappings in both p-vector spaces (including locally p-convex spaces); 2) the general principle of Leray-Schauder alternatives; and 3) various forms of fixed point theorems for non-self mappings.Specifically, the book focuses on the development of general fixed point theory for both single and set-valued mappings. It provides affirmative answers to the Schauder conjecture under the general setting of p-vector spaces (including topological vector spaces as a special class), and locally p-convex spaces. The book establishes best approximation results for upper semi-continuous and 1-set contractive set-valued mappings, which are used as tools to establish new fixed point theorems for non-self set-valued mappings with either inward or outward set conditions under various situations. These results improve or unify corresponding results in the existing literature for nonlinear analysis, and allow for the establishment of the fundamental general theory for the development of fixed point theorems in topological vector spaces since Schauder's conjecture was raised in 1930.This new book is a staple textbook for undergraduate and postgraduate students, a reference book for researchers in the field of fixed point theory in nonlinear functional analysis, and is easily accessible to general readers in mathematics and related disciplines.