  Discrete point sets $\mathcal{S}$ such as lattices or quasiperiodic Delone
sets may permit, beyond their symmetries, certain isometries $R$ such that
$\mathcal{S}\cap R\mathcal{S}$ is a subset of $\mathcal{S}$ of finite density.
These are the so-called coincidence isometrie. They are important in
understanding and classifying grain boundaries and twins in crystals and
quasicrystals. It is the purpose of this contribution to introduce the
corresponding coincidence problem in a mathematical setting and to demonstrate
how it can be solved algebraically in dimensions 2, 3 and 4. Various examples
both from crystals and quasicrystals are treated explicitly, in particular
(hyper-)cubic lattices and quasicrystals with non-crystallographic point groups
of type $H_2$, $H_3$ and $H_4$. We derive parametrizations of all linear
coincidence isometries, determine the corresponding coincidence index (the
reciprocal of the density of coinciding points, also called
$\varSigma$-factor), and finally encapsulate their statistics in suitable
Dirichlet series generating functions.
