  The classical approximation provides a non-perturbative approach to
time-dependent problems in finite temperature field theory. We study the
divergences in hot classical field theory perturbatively. At one-loop, we show
that the linear divergences are completely determined by the classical
equivalent of the hard thermal loops in hot quantum field theories, and that
logarithmic divergences are absent. To deal with higher-loop diagrams, we
present a general argument that the superficial degree of divergence of
classical vertex functions decreases by one with each additional loop: one-loop
contributions are superficially linearly divergent, two-loop contributions are
superficially logarithmically divergent, and three- and higher-loop
contributions are superficially finite. We verify this for two-loop SU(N)
self-energy diagrams in Feynman and Coulomb gauges. We argue that hot,
classical scalar field theory may be completely renormalized by local (mass)
counterterms, and discuss renormalization of SU(N) gauge theories.
