  Point-vortex dynamics describe idealized, non-smooth solutions to the
incompressible Euler equations on 2-dimensional manifolds. Integrability
results for few point-vortices on various domains is a vivid topic, with many
results and techniques scattered in the literature. Here we give a unified
framework for proving integrability results for $N=2$, $3$, or $4$
point-vortices (and also more general Hamiltonian systems), based on symplectic
reduction theory. The approach works on any 2-dimensional manifold; we
illustrate it on the sphere, the plane, the hyperbolic plane, and the flat
torus. A systematic study of integrability is prompted by advances in
2-dimensional turbulence, bridging the long-time behaviour of 2D Euler
equations with questions of point-vortex integrability. A gallery of solutions
is given in the appendix.
