As the author says in the Introduction, it is impossible to cover the whole of convexity in one textbook. Nevertheless, this book shows that basic principles can be presented in an accessible way and demonstrated on various examples and applications. After introductory results on general convex sets in Euclidean spaces and general topological vector spaces (theorems of Caratheorory, Helly and Radon, separation results and the Krein-Milman theorem), the author goes to applications in discrete and combinatorial convexity. The role of duality in convexity is illustrated on examples. The further topics are the ellipsoidal approximation, combinatorial structure of polytopes and interaction of convex bodies with lattices. The applications range over different areas, including analysis, probability, algebra, combinatorics and number theory. The presentation is very clear and suitably chosen applications illustrate abstract results and make the reading more attractive.