Quantum maps are presented with special emphasis on their physical origin. They represent a testing ground for understanding concepts in quantized chaotic systems. The book teaches the modern mathematical methods from analytic and algebraic number theory as applied to quantum maps. It gives a broad and in depth overview of the mathematical problems arising in this area. Also treated are the numerical aspects in quantum chaos such as eigenvalue and eigenfunctions computations for chaotic quantum systems. The book addresses scientists and advanced students in mathematics and mathematical physics.