Main papers on quantum games are written by physicists for physicists, and the inevitable exploitation of physics jargon may create difficulties for mathematicians or economists. Our goal here is to make clear the physical content and to stress the new features of the games that may be revealed in their quantum versions. We first introduce from scratch the most fundamental facts of finite-dimensional quantum mechanics. In the main sections the foundations of quantum games are built via the basic examples. We omit sometimes the lengthy calculations (referring to the original papers) once the physical part is sorted out and the problem is reformulated as pure game-theoretic problem of calculating the Nash or dominated equilibria. Finally we touch upon general theory of finite quantum static games and provide further links and references. Quantum games: a survey for mathematicians