Discrete Morse Homology and the Perseus Software Project. At its core, the computation of homology -- persistent or otherwise -- involves performing elementary row and column operations on possibly gigantic matrix representations of boundary operators defined on cell complexes. Discrete Morse theory provides a powerful and flexible framework for drastically reducing the sizes of these matrices in almost linear time while preserving all the underlying homological information. In this talk, we outline the scope and limitations of such Morse-theoretic methods. We also provide a brief overview of the Perseus software project which implements these methods in order to efficiently compute persistent homology of various types of filtered complexes.