  We prove the existence of periodic tessellations of $\mathbb{R}^N$ minimizing
a general nonlocal perimeter functional, defined as the interaction between a
set and its complement through a nonnegative kernel, which we assume to be
either integrable at the origin, or singular, with a fractional type
singularity. We reformulate the optimal partition problem as an isoperimetric
problem among fundamental domains associated with discrete subgroups of
$\mathbb{R}^N$ , and we provide the existence of a solution by using suitable
concentrated compactness type arguments and compactness results for lattices.
Finally, we discuss the possible optimality of the hexagonal tessellation in
the planar case.
