  We consider the problem of counting and classifying domino tilings of a
quadriculated torus. The counting problem for rectangles was studied by
Kasteleyn and we use many of his ideas. Domino tilings of planar regions can be
represented by height functions; for a torus given by a lattice L, these
functions exhibit arithmetic L-quasiperiodicity. The additive constants
determine the flux of the tiling, which can be interpreted as a vector in the
dual lattice (2L)*. We give a characterization of the actual flux values, and
of how corresponding tilings behave. We also consider domino tilings of the
infinite square lattice; tilings of tori can be seen as a particular case of
those. We describe the construction and usage of Kasteleyn matrices in the
counting problem, and how they can be applied to count tilings with prescribed
flux values. Finally, we study the limit distribution of the number of tilings
with a given flux value as a uniform scaling dilates the lattice L.
