  We investigate the dynamics of $N$ point vortices in the plane, in the limit
of large $N$. We consider {\em relative equilibria}, which are rigidly rotating
lattice-like configurations of vortices. These configurations were observed in
several recent experiments [Durkin and Fajans, Phys. Fluids (2000) 12, 289-293;
Grzybowski {\em et.al} PRE (2001)64, 011603]. We show that these solutions and
their stability are fully characterized via a related {\em aggregation model}
which was recently investigated in the context of biological swarms [Fetecau
{\em et.al.}, Nonlinearity (2011) 2681; Bertozzi {\em et.al.}, M3AS (2011)]. By
utilizing this connection, we give explicit analytic formulae for many of the
configurations that have been observed experimentally. These include
configurations of vortices of equal strength; the $N+1$ configurations of $N$
vortices of equal strength and one vortex of much higher strength; and more
generally, $N+K$ configurations. We also give examples of configurations that
have not been studied experimentally, including $N+2$ configurations where $N$
vortices aggregate inside an ellipse. Finally, we introduce an artificial
``damping'' to the vortex dynamics, in an attempt to explain the phenomenon of
crystalization that is often observed in real experiments. The diffusion breaks
the conservative structure of vortex dynamics so that any initial conditions
converge to the lattice-like relative equilibrium.
