This paper describes the latest version (3.x) of the Maude-NRL Protocol Analyzer (Maude-NPA), a tool for the analysis of cryptographic protocols that provides support for equational theories. A key feature is it supports reasoning about a class of theories with disjoint compositions that have the finite variant property, via the use of a unification algorithm called variant unification. These theories include exclusive-or, Abelian groups, and theories underlying Diffie-Hellman key exchange. In this paper we show how Maude-NPA works, and we explain in detail the variant unification algorithm it uses and how the tool makes use of it. We also describe work on another form of unification, asymmetric unification, that has the potential to improve on the performance of variant unification in some cases.