In this chapter, the authors explain a new homology theory. They describe the singular homology groups and prove that they satisfy the Eilenberg-Steenrod axioms on the class of all topological spaces. However, the singular homology groups are not immediately computable. One must develop a good deal of singular theory before one can compute the homology of even such a simple space as the sphere. The authors provide the Jordan curve theorem, theorems about manifolds, and the computation of the homology of real and complex projective spaces. Since inclusion induces an isomorphism of the absolute homology groups of the respective sequences, the Five-lemma implies that it induces an isomorphism of the relative groups as well. The authors state that one of the advantages of simplicial homology theory is its effective computability. They refine the ad hoc techniques into a systematic method for computing homology groups.