We study to what extent the three-dimensional SU( N) + adjoint Higgs theory can be used as an effective theory for finite-temperature SU( N) gauge theory, with N = 2, 3. The parameters of the 3d theory are computed in 2-loop perturbation theory in terms of T/Λ overlineMS, N and N f. The perturbative effective potential of the 3d theory is computed to two loops for N = 2. While the Z(N) symmetry probably driving the 4d confinement-deconfinement phase transition (for Nf = 0) is not explicit in the effective Lagrangian, it is partly reinstated by radiative effects in the 3d theory. Lattice simulations in the 3d theory are carried out for N = 2, and the static screening masses relevant for the high-temperature phase of the 4d theory are measured. In particular, we measure non-perturbatively the O( g2T) correction to the Debye screening mass. We find that non-perturbative effects are much larger in the SU(2) + adjoint Higgs theory than in the SU(2) + fundamental Higgs theory.