Given a Banach space and a bounded linear operator on it, the invariant subspace problem refers to the existence of a proper subspace invariant under the operator. We discuss a history of this problem and discuss a solution in the case of subalgebras of bounded operators on a Banach space that contain compact operators. The highlight is on the use of fixed point theorems for the same. Speaker(s): Malavika Mukundan (University of Michigan)