  We review the status of non-perturbative lattice studies of the electroweak
phase transition. In the Standard Model, the complete phase diagram has been
reliably determined, and the conclusion is that there is no phase transition at
all for the experimentally allowed Higgs masses. In the Minimal Supersymmetric
Standard Model (MSSM), in contrast, there can be a strong first order
transition allowing for baryogenesis. Finally, we point out possibilities for
future simulations, such as the problem of CP-violation at the MSSM electroweak
phase boundary.
