We investigate in a systematic way hypercontractivity property in Orlicz spaces for Markov semi-groups related to homogeneous and non homogeneous diffusions in $\mathbb{R}^n$. We provide an explicit construction of a family of Orlicz functions for which we prove that the associated hypercontractivity property is equivalent to a suitable functional inequality. Connections between our results and the Talagrand Conjecture on the regularization effect of the Ornstein-Uhlenbeck semi-group in $\mathbb{L}_1$ are also discussed.