  We study random normal matrix models whose eigenvalues tend to be distributed
within a narrow "band" around the unit circle of width proportional to
$\frac1n$, where $n$ is the size of matrices. For general radially symmetric
potentials with various boundary conditions, we derive the scaling limits of
the correlation functions, some of which appear in the previous literature
notably in the context of almost-Hermitian random matrices. We also obtain that
fluctuations of the maximal and minimal modulus of the ensembles follow the
Gumbel or exponential law depending on the boundary conditions.
