Hyperbolicity is a central notion in the study of chaotic dynamical systems. Unfortunately, expansivity which is one if its main ingredients is very uncommon in typical dynamics. Because of this limitation, over the years some generalizations appeared in the literature, trying to preserve main features of hyperbolic, yet present in much more generality. In 1993 Kato introduced the notion of continuum-wise expansive homeomorphisms, and in 2024 it was used by Artigue, Carvalho, Cordeiro and Vieitez to define continuum-wise hyperbolicity. This definition combines cw-expansive with kind of local product structure, also expressed in terms of evolution of continua. In this talk we will survey selected results for surface dynamics and present new results obtained by the author jointly with several collaborators.