The static sectors of the electroweak Standard Model and QCD at finite temperatures are described by 3d SU(N) Higgs models with scalars in the fundamental and adjoint representation, respectively. I summarize the non-perturbative physics of these theories like mass spectrum, string tension, screening of the static potential and non-perturbative corrections to the Debye mass, as obtianed from recent lattice simulations. AIt is observed that in the 3d theory corresponding to the purely magnetic sector there is a larger hierarchy between different quanftities than between the elecftric (approx gT) and magnetic (approx g~2T) sectors of finite T gauge theories, which are not generally well separated.
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