We will discuss some super-rigidity results in the ”higher-rank to hyperbolic-like target” paradigm, and two applications of those: one to Monod-Shalom type rigidity for products of convergence groups, and another to a geometric question of classifying hyperbolic structures on some groups. Based on joint works with U.Bader, and another one with U.Bader. P.-E. Caprace, A. Sisto.