In this talk I plan to discuss examples and the dynamics of topologically hyperbolic homeomorphisms and recent generalizations considered in joint works with Welington Cordeiro (IMPAN), Alfonso Artigue and José Vieitez (UDELAR).
As an attempt to classify the cw-expansive homeomorphisms satisfying the shadowing property we introduce cw-hyperbolicity generalizing the usual local product structure of the hyperbolic scenario. We prove a shadowing lemma in the cw-hyperbolic case in the form of the L-shadowing property and obtain consequences on the dynamics of such systems. We classify cw-hyperbolic homeomorphisms with respect to the number of intersections between local stable and unstable continua through the notion of cwN-hyperbolicity and prove these share additional properties with the hyperbolic systems.