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The quantum cat maps are the Weil represention of a subgroup of SL(2,Z). They are among the most important models used to study the quantum mechanical properties of dynamical systems whose corresponding classical dynamics is hyperbolic. In this talk the main features of these systems will be discussed. In particular for each quantum map U(A) there exists a set of unitary operators that commute with U(N) and among themselves. Such operators (known as Hecke operators in analogy with an analogous phenomenon in the theory of modular surfaces) are responsible for the unusual properties of these systems, like their spectral degeneracy. We shall also describe how these maps can be coupled with SU(2) matrices to study quantum system with spin when the underlying dynamics is chaotic, for which relatively few models are available.