The problem of time is a central feature of quantum cosmology: differing from ordinary quantum mechanics, in cosmology there is nothing outside the system which plays the role of clock, and this makes difficult the obtention of a consistent quantization. A possible solution is to assume that a subset of the variables describing the state of the universe can be a clock for the remaining of the system. Following this line, in this book a new proposal consisting in the previous identification of time by means of gauge fixation is applied to the quantization of homogeneous cosmological models. Both path integral and canonical formulations are studied for relativistic and string cosmologies; in particular, a complete chapter about low energy string cosmology is included. The required basic concepts (as for example the Hamiltonian formulation of General Relativity) are reviewed, so that the book can be of interest not only for a researcher but also for a student.