In this thesis we study aspects of the cosmological electroweak phase transition which are relevant to the possibility of baryogenesis at this epoch. We focus on two issues: first, requiring that the observed baryon number be of electroweak origin places strong constraints on electroweak physics, and second, baryogenesis at the electroweak scale may be driven by an asymmetry generated at the GUT scale. We use the effective potential at finite temperature as a means of analyzing phase transitions associated with spontaneous symmetry breaking. We develop the theory with two basic examples: the scalar and Abelian Higgs models. Infrared divergences near the phase transition make the one-loop description unreliable, and indeed invalidate conventional perturbation theory. Borrowing a method from studies of QCD at high temperatures, we demonstrate that the summation of ring diagrams cures the leading infrared divergences and achieves a more reliable perturbative expansion. We then apply this formalism to the minimal Standard Model, following previous work, and confirm weak first-order behavior at the phase transition. We show that requiring the baryon number not be erased by sphaleron processes after the phase transition places a stringent bound on the Higgs mass, which is incompatible with experiment. This cosmological bound, however, may be relaxed by extending the scalar sector of the Standard Model. We consider the two simplest such extensions, the addition of a gauge singlet and of a second doublet. We demonstrate that ring-improvement in the singlet extension alters previous arguments at the one-loop level and yields a more restrictive bound on the Higgs mass. While ring-improvement in the two-doublet model, in principle, also reduces the Higgs mass bound found earlier at one loop, the multitude of new couplings in this model does not permit a definitive statement. We then investigate a mechanism for generating the observed baryon asymmetry (n_B/S~ 10-¹⁰) at the electroweak phase transition from a pre-existing leptonic asymmetry (L_T/s~ 10-⁵) produced at the GUT scale. This mechanism works by charge transport in a strongly first-order phase transition and avoids the need for large CP-violation at the electroweak scale.