This book describes and gives applications of an important new tool in the study of complex analytic hypersurface singularities: the Lˆ cycles of the hypersurface. The Lˆ cycles and their multiplicities - the Lˆ numbers - provide effectively calculable data which generalizes the Milnor number of an isolated singularity to the case of singularities of arbitrary dimension. The Lˆ numbers control many topological and geometric properties of such non-isolated hypersurface singularities. This book is intended for graduate students and researchers interested in complex analytic singularities.