New homology groups are defined for a non-self-adjoint operator algebra with a distinguished masa which is based upon cycles and boundaries associated with complexes of partial isometries in the stable algebra. Under natural hypotheses the zeroth order group coincides with theK0group of the generatedC*-algebra. Several identifications and applications are given, and in particular it is shown how stable homology is significant for the classification of regular subalgebras and regular limit algebras.