We generalize the usual notion of topological hyperbolicity introducing the concept of continuum-wise hyperbolicity. We discuss examples of these systems and characterize the possible dynamical phenomena that can occur on cw-hyperbolic transitive homeomorphisms: either they are topologically hyperbolic, or there exist arbitrarily small dynamical balls containing topological semi-horseshoes, that are periodic sets where the dynamics is semiconjugated to the shift of two symbols. We prove cw-hyperbolicity implies some of the standard properties of hyperbolic systems, such as the shadowing property and finiteness of chain recurrent classes. Work in progress with Alfonso Artigue, Bernardo Carvalho and Jose Vieitez.