We discuss the formulation of "thermal renormalization group-equations" and
their application to the finite temperature phase-transition of scalar
O(N)-theories. Thermal renormalization group-equations allow for a computation
of both the universal and the non-universal aspects of the critical behavior
directly in terms of the zero-temperature physical couplings. They provide a
nonperturbative method for a computation of quantities like real-time
correlation functions in a thermal environment, where in many situations
straightforward perturbation theory fails due to the bad infrared-behavior of
the thermal fluctuations. We present results for the critical temperature,
critical exponents and amplitudes as well as the scaling equation of state for
self-interacting scalar theories.Comment: 32 pages with 10 figures and 4 tables included, latex2
