The purpose of this paper is twofold.  On one hand we reconsider the continuous time limit of the GARCH(1,1) process and show that, by choosing diffferent reparameterizations, as a function of the discrete interval h, we obtain, as h -> 0, either anon-degenerate or a degenerate diffusion limit.  On the other hand, we introduce a class of GARCH-type processes, that we call "modified semi-strong GARCH" and derive their continuous time limit. Special attention is paid to the analysis of the support of the transition function of the different diffusion limits we obtain.