  We investigate the existence of collision-free nonconstant periodic solutions
of the $N$-vortex problem in domains $\Omega\subset\mathbb{C}$. These are
solutions $z(t)=(z_1(t),\dots,z_N(t))$ of the first order Hamiltonian system \[
\dot{z}_k(t)=-i\nabla_{z_k} H_\Omega\big(z(t)\big),\quad k=1,\dots,N, \] where
the Hamiltonian $H_\Omega$ has the form \[ H_\Omega(z_1,\dots,z_N)
  = \frac1{2\pi}\sum_{{j,k=1}\atop{j\ne k}}^N \log\frac1{|z_j-z_k|} - F(z). \]
The function $F:\Omega^N\to\mathbb{R}$ depends on the regular part of the
hydrodynamic Green's function and is unbounded from above. The Hamiltonian is
unbounded from above and below, it is singular, not integrable, energy surfaces
are not compact and not known to be of contact type. We prove the existence of
a family of periodic solutions $z^r(t)$, $0<r<r_0$, with arbitrarily small
minimal period $T_r\to0$ as $r\to0$. The solutions are close to the singular
set of $H_\Omega$. Our result applies in particular to generic bounded domains,
which may be simply or multiply connected. It also applies to certain unbounded
domains. Depending on the domain there are multiple such families.
