This is a brief, but very informative survey article of the basic results concerning the question of existence of fixed points of nonexpansive mappings in Banach spaces. The structure of this article is the following: Introduction. Principal notations. Chapter 1: Normal structure in Banach spaces and its generalizations; 1.1. Normal structure; 1.2. Uniformly normal structure; 1.3. Close-to-normal structure; 1.4. Weakly normal structures; 1.5. Fixed points and product spaces. Chapter 2: The method of asymptotic center; 2.1. Asymptotic centers of bounded sequences; 2.2. Multi-valued mappings; 2.3. Uniformly Lipschitzian mappings. Chapter 3: Nonstandard methods in the fixed point theory; 3.1. The example of Alspach; 3.2. Ultrapower and superreflexivity; 3.3. Ultrapower and fixed points. Some unsolved problems. Literature cited (228 articles). In Chapter 3, the author, using the nonstandard method, gives a positive answer to a question of \textit{D. Amir} [Pac. J. Math. 118, 1-15 (1985; Zbl 0557.46011)]: every Banach space with uniformly normal structure is superreflexive. As supplement we refer to: Recent advances on metric fixed point theory (\textit{T. Dominguez Benavides}, ed.), Univ. de Sevilla (1996; Zbl 0859.00024).