The study of stratified spaces is concerned with topological spaces that can be decomposed nicely into more familiar spaces, typically manifolds. The classical theory is concerned soley with spaces whose strata are finite-dimensional manifolds. Infinite-dimensional spaces arise naturally even in the study of finite-dimensional spaces and are fundamental to modern physics. However, the notion of stratification of a space into such infinite-dimensional spaces has only been studied incidentally. Here we will extend much of the theory of finite-dimensional stratifications to allow for this. In the finite-dimensional case, the space can be seen as being glued together along the strata and this data can be used to glue together sheaves on the strata into sheaves on the original space. We will extend this notion to consider gluing together bundles more general than the étalé bundles that correspond to sheaves. This allows for a reformulation of the Whitney A-regularity condition that guarantees the existence of a stratified tangent bundle on classical stratified spaces. In the infinite-dimensional case, the space is not necessarily given by gluing together the strata in the same way. When it is, we can extend Artin gluing of bundles to this context as well and use the reformulated A-regularity condition to construct stratified tangent bundles for these infinite-dimensional manifolds. Finally, we will apply these results to problems in mathematical physics. In particular, we will stratify and construct stratified tangent bundles on the state spaces of a C^∗-algebra, corresponding to the states of a quantum system.