We will explore the L-shadowing property. First of all we will show basic facts about this property and compare it with the topological hyperbolicity. We will show that systems with expansivity and the shadowing property satisfy the L-shadowing property, and that there are examples with the L-shadowing property which are not expansive. Then we will prove a spectrum decomposition theorem for system with the L-shadowing and generalize the usual notion of topological hyperbolicity introducing the concept continuum-wise hyperbolicity.