The Invariant Subspace Problem is one of the most famous problems in Operator Theory, and is concerned with the search of non-trivial, closed, invariant subspaces for bounded operators acting on a separable Banach space. Considerable success has been achieved over the years both for the existence of such subspaces for many classes of operators, as well as for non-existence of invariant subspaces for particular examples of operators. However, for the most important case of a separable Hilbert space, the problem is still open. \\ \\ A natural, related question deals with the existence of invariant subspaces for perturbations of bounded operators. These types of problems have been studied for a long time, mostly in the Hilbert space setting. In this talk I will present a new approach to these ``perturbation'' questions, in the more general setting of a separable Banach space. I will focus on the recent history, presenting several new results that were obtained along the way with this new approach, and examining their connection and relevance to the Invariant Subspace Problem.