  Perturbation theory alone fails to describe thermodynamics of the electroweak
phase transition. We review a technique combining perturbative and
non-perturbative methods to overcome this challenge. Accordingly, the principal
theme is a tutorial of high-temperature dimensional reduction. We present an
explicit derivation with a real singlet scalar and compute the thermal
effective potential at two-loop order. In particular, we detail the dimensional
reduction for a real-singlet extended Standard Model. The resulting effective
theory will impact future non-perturbative studies based on lattice simulations
as well as purely perturbative investigations.
