We consider a purely variational approach to time dependent problems, yielding the existence of global parabolic minimizers. These evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals. In the simplest situation, the method provides the existence of global weak solutions to Cauchy-Dirichlet problems of parabolic systems. This is a joint collaboration with V. B?gelein and F. Duzaar.