In this thesis the close relationship between the topological K-homology group of the spacetime manifold X of string theory and D-branes in string theory is examined. An element of the K-homology group is given by an equivalence class of K-cycles [M,E,ϕ], where M is a closed spinc manifold, E is a complex vector bundle over M, and ϕ:M→X is a continuous map. It is proposed that a K-cycle [M,E,ϕ] represents a D-brane configuration wrapping the subspace ϕ(M). As a consequence, the K-homology element defined by [M,E,ϕ] represents a class of D-brane configurations that have the same physical charge. Furthermore, the K-cycle representation of D-branes resembles the modern way of characterizing fundamental strings, in which the strings are represented as two-dimensional surfaces with maps into the spacetime manifold. This classification of D-branes also suggests the possibility of physically interpreting D-branes wrapping singular subspaces of spacetime, enlarging the known types of singularities that string theory can cope with.