String theory is a a physical field theory in which point particles are replaced by 1-manifolds propagating in time, called strings. The 2-manifold representing the time evolution of a string is called the string worldsheet. Strings can be either closed (meaning their worldsheets are closed surfaces) or open (meaning their worldsheets have boundary). A D-brane is a submanifold of the spacetime manifold on which string endpoints are constrained to lie.

There are five different string theories that have supersymmetry, and they are all related by various dualities. This dissertation will review how D-branes are classified by K-theory. We will then explore the $K$-theoretic aspects of a hypothesized duality between the type I theory compactified on a 4-torus and the type IIA theory compactified on a K3 surface, by looking at a certain blow down of the singular limit of K3. This dissertation concludes by classifying D-branes on the type II orientifold T⁴/Z₂ when the Z₂ action is multiplication by -1 and the H-flux is trivial. We find that classifying D-branes on the singular limit of K3, T⁴/Z₂ by equivariant K-theory agrees with the classification of D-branes on a smooth K3 surface by ordinary K-theory.