This introduction to the theory of metric spaces carries the student through the motivation, development, and applicationof an abstract mathematical concept. Yu. A. Shreider begins with a definition of mathematical concepts and then, to motivate his definition of “ metric space,” the author evaluates distance functions and their properties in elementary geometry. After introducing two sets of metric space axioms, he surveys the properties of open balls, sequences, Cauchy sequences, complete metric spaces, and isometries. In the following chapter, the author offers a number of informative examples.Professor Shreider turns next to applications of the theories, especially the theory of coding and cybernetics. Then returning to theory, the author defines an n-dimensional vector space over real numbers and develops the theory of norms and metricson Euclidean n-space. The final chapter considers the possibility of generalizing the concept of metric space.