We consider the solution of parabolic partial differential equations. In standard time- stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time-step. By contrast, the method presented in this paper treats the whole of the space-time problem simultaneously. Thus the multigrid operations of smoothing and coarse grid correction are defined on all of the space-time variables of a given grid-level. The method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil. Numerical results for the one- and two-dimensional heat equation are presented and are shown to agree closely with predictions from Fourier mode analysis.