The special seminar is dedicated to the current state of the theory of quantum computing. We will briefly consider the physical foundations of the processes that occur in quantum computing and, rather quickly, reduce them to linear operators in finite-dimensional spaces. After this, we consider the theory of languages and computability on quantum computers. The main part of the seminar will be devoted to the study of various algorithms on quantum computers. We conclude the seminar with a discussion of recent progress in building physical devices that implement various models of quantum computing.