  We study the high-temperature phase transitions for the Abelian and $SU(2)$
Higgs models, using the exact renormalization group. The evolution equation for
a properly-defined coarse-grained free energy is solved. The phase diagram of
the Abelian Higgs model has a region of second-order phase transitions governed
by fixed points, for which we calculate critical exponents and crossover
curves. It also has a region of first-order phase transitions, which we discuss
in terms of the coarse-grained free energy. The $SU(2)$ Higgs model for small
Higgs field masses has only first-order phase transitions. We derive their
properties and compare with lattice and perturbative results. For larger Higgs
field masses the strength of the first-order phase transitions diminishes. For
masses larger than 80-100 GeV we find evidence that they are replaced by
analytical crossovers.
