We develop the basic properties of PL and singular intersection homology. This includes the behavior of the intersection homology groups under stratified maps and homotopies and the invariance of intersection homology groups under stratified homotopy equivalences. We introduce relative intersection homology, the long exact sequence of a pair, Mayer–Vietoris sequences, and excision. An important special computation is that of the intersection homology of a cone, which provides a good basic example of an intersection homology computation but also provides a formula that plays an essential role throughout the theory, as all points in pseudomanifolds have neighborhoods that are stratified homotopy equivalent to cones.