Especially since the 1997 Nobel Prize winning work of S. Chu, C. Cohen-Tannoudji, and W. Phillips on Bose-Einstein condensates (BECs), there has been a great deal of exciting experimental and theoretical work in the area. Recently, there has been intriguing experimental and theoretical work on the creation and dynamics of vortices in one- and two-species condensates. The governing equations for these systems are Hamiltonian; in fact, they are simply nonlinear Schrodinger equations with nonhomogeneous terms. In addition to questions of existence and stability of these vortices, which are of interest in their own right, the analysis leads to several intriguing fundamental mathematical problems for Hamiltonian systems. These include (a) the connection between integrability and a linear stability analysis, (b) the role in which the energy plays in a stability analysis, and (c) the dynamical effect caused by the wave not being a minimizer for the energy. Answers to some of these problems will be presented, in general and in the context of BECs, as well as interesting open questions.