A stratified space is a kind of topological space together with a partition into smooth manifolds. These kinds of spaces naturally arise in the study of singular algebraic varieties, symplectic reduction, and differentiable stacks. In this paper, we introduce a particular class of stratified spaces called stratified vector bundles, and provide an alternate characterization in terms of monoid actions. We will then provide large families of examples coming from the theory of Whitney stratified spaces, singular foliation theory, and equivariant vector bundle theory. Finally, we extend functorial properties of smooth vector bundles to the stratified case.