In the last decades, advances in pulse shaping techniques have opened up the possibility of manipulation of systems whose evolution follows the laws of quantum mechanics. Moreover, novel applications, such as in quantum information processing, have offered further motivation for this study. From a mathematical point of view, the field which is now known as'Quantum Control' is a combination of different mathematical techniques borrowed from a wide variety of mathematical areas. Different tools apply to different models which correspond to different approximations of the physical system at hand. The simplest case is the one of a closed system, i.e., a system non interacting with the environment in any way other than through the external controls, controlled in open loop, and whose state can be modeled as a vector in a finite dimensional Hilbert space. In this case, the operator describing the evolution belongs to a Lie group and the control system is determined by a family of right invariant vector fields on such a Lie group. Techniques of geometric control are therefore appropriate. As some of the above assumptions on the physical model are relaxed, different tools have to be used. The consideration of 'open' systems, which also allow for a continuous measurement of the state and feedback, requires the introduction of techniques of dynamical semigroups as well as stochastic calculus. The study of infinite dimensional quantum control systems is often done using tools of functional analysis and control of partial differential equations. This talk is a brief survey of the field from the point of view of the mathematics that is used and needs to be developed. After introducing basic notions of quantum mechanics and the relevant models used in applications I will indicate a number of open mathematical problems.