The purpose of the book is to give a comprehensive and detailed introduction to the computational complexity of counting and sampling. It starts with the basics (from the algorithmic point of view: dynamic programming algorithms; from the theoretical computer science point of view: P, NP, NP-complete) and aims to the advanced topics like stochastic approximations of counting combinatorial objects and holographic algorithms. The book will consist of three main topics: I. Counting problems that are solvable in polynomial running time, II. Approximation algorithms for counting and sampling, III. Holographic algorithms.