  We study the finite temperature electroweak phase transition with lattice
perturbation theory and Monte Carlo techniques. Dimensional reduction is used
to approximate the full four-dimensional SU(2) + a fundamental doublet Higgs
theory by an effective three-dimensional SU(2) + adjoint Higgs + fundamental
Higgs theory with coefficients depending on temperature via screening masses
and mass counterterms. Fermions contribute to the effective theory only via the
$N_F$ and $m_{\rm top}$ dependence of the coefficients. For sufficiently small
lattices ($N^3 < 30^3$ for $m_H$ = 35 GeV) the study of the one-loop lattice
effective potential shows the existence of the {\em second} order phase
transition even for the small Higgs masses. At the same time, a clear signal of
a {\em first order} phase transition is seen on the lattice simulations with a
transition temperature close to but less than the value determined from the
perturbative calculations. This indicates that the dynamics of the first order
electroweak phase transition depends strongly on non-perturbative effects and
is not exclusively related to the so-called $\phi^3$ term in the effective
potential.
