We introduce a new class of convex-regularized Optimal Transport losses, which generalizes the classical Entropy-regularization of Optimal Transport and Sinkhorn divergences, and propose a generalized Sinkhorn algorithm. Our framework unifies many regularizations and numerical methods previously appeared in the literature. We show the existence of the maximizer for the dual problem, complementary slackness conditions, providing a complete characterization of solutions for such class of variational problems. As a consequence, we study structural properties of these losses, including continuity, differentiability and provide explicit formulas for its gradient. Finally, we provide theoretical guarantees of convergences and stability of the generalized Sinkhorn algorithm, even in the continuous setting. The techniques developed here are directly applicable also to study Wasserstein barycenters or, more generally, multi-marginal problems.