Abstract: Among researchers in rigidity theory, there has been a growing feeling over the past few years that quite general analogues of the amazing homomorphism extension theorems for lattices in (higher-rank) semisimple Lie groups, the superrigidity theorems, should exist in the presence of appropriate "negative curvature" assumptions. A number of such results are now known. In this talk, a (commensurator) superrigidity theorem will be proven for homomorphisms of lattices in arbitrary locally compact second countable groups under the very mild negative curvature assumption that the target group admit a convergence group action. This significant improvement on earlier results in this direction applies, for example, when the target is a quasiconformal group of a uniform domain, a quasi-isometry group of a proper, geodesic Gromov hyperbolic metric space, or a uniformly quasi-mobius group acting on a compact metrizable space. Those with an interest in rigidity theory and/or metric space and group theoretic manifestations of negative curvature phenomena are especially encouraged to attend.