  We provide a framework to build periodic boundary conditions on the
pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian
space of constant negative curvature. Starting from the common case of periodic
boundary conditions in the Euclidean plane, we introduce all the needed
mathematical notions and sketch a classification of periodic boundary
conditions on the hyperbolic plane. We stress the possible applications in
statistical mechanics for studying the bulk behavior of physical systems and we
illustrate how to implement such periodic boundary conditions in two examples,
the dynamics of particles on the pseudosphere and the study of classical spins
on hyperbolic lattices.
