The book is a concise, self-contained and up-to-date introduction to extremal combinatorics for non-specialists. Strong emphasis is made on theorems with particularly elegant and informative proofs which may be called gems of the theory. A wide spectrum of most powerful combinatorial tools is presented: methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A throughout discussion of some recent applications to computer science motivates the liveliness and inherent usefulness of these methods to approach problems outside combinatorics. No special combinatorial or algebraic background is assumed. All necessary elements of linear algebra and discrete probability are introduced before their combinatorial applications. Aimed primarily as an introductory text for graduates, it provides also a compact source of modern extremal combinatorics for researchers in computer science and other fields of discrete mathematics.