We study the possibility of relaxing the cosmological bound on the Higgs mass coming from the requirement of non-erasure of the baryon asymmetry by sphalerons. After reviewing the Standard Model case we obtain this bound in two extensions of it: 1) The Standard Model with an additional gauge singlet, and 2) The Minimal Supersymmetric Standard Model. Taking fully into account all experimental constraints and thermal screening effects we found that the situation can be slightly improved with respect to the Standard Model but only in case 1) a non negligible region in parameter space exists where the baryon washout is avoided and the experimental bounds evaded. (Talk given at the NATO Advanced Reseach Workshop: "Electroweak Physics and the early Universe", 23-25 March 1994, Sintra (Portugal).)