We investigate the weak-coupling expansion at high temperature for relativistic quantum field theories. Boson self-energies generally acquire a non-analytic dependence on the coupling due to the infrared plasmon effect. We show how to sum loops within loops for such thermodynamic quantities as pressure and energy density. Recent high-precision Monte Carlo simulations of quarkless SU(3) gauge theory are compared to the weak-coupling expansion and an excellent agreement is obtained for the high-temperature phase. Weak-coupling calculations are also carried out for supersymmetric grand unified models. We find indications that such models are often strongly coupled already at the unification temperature. This may have a significant effect on models of the early universe.