This book is an introduction to convex analysis and some of its applications. It starts with basic theory, which is explained within the framework of finite-dimensional spaces. The only prerequisites are basic analysis and simple geometry. The second chapter presents some applications of convex analysis, including problems of linear programming, geometry, and approximation. Special attention is paid to applications of convex analysis to Kolmogorov-type inequalities for derivatives of functions in one variable. Chapter 3 collects some results on geometry and convex analysis in infinite-dimensional spaces. A comprehensive introduction written 'for beginners' illustrates the fundamentals of convex analysis in finite-dimensional spaces. The book can be used for an advanced undergraduate or graduate-level course on convex analysis and its applications. It is also suitable for independent study of this important area of mathematics.