In this thesis the close relationship between the topological -homology group of the spacetime manifold of string theory and D-branes in string theory is examined. An element of the -homology group is given by an equivalence class of -cycles , where is a closed spin manifold, is a complex vector bundle over , and is a continuous map. It is proposed that a -cycle represents a D-brane configuration wrapping the subspace . As a consequence, the -homology element defined by represents a class of D-brane configurations that have the same physical charge. Furthermore, the -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.