  We present locally stabilized, conforming space-time finite element methods
for parabolic evolution equations on hexahedral decompositions of the
space-time cylinder. Tensor-product decompositions allow for anisotropic a
priori error estimates, that are explicit in spatial and temporal meshsizes.
Moreover, tensor-product finite elements are suitable for anisotropic adaptive
mesh refinement strategies provided that an appropriate a posteriori
discretization error estimator is available. We present such anisotropic
adaptive strategies together with numerical experiments.
