  We study square-tiled tori, that is, tori obtained from a finite collection
of unit squares by parallel side identifications. Square-tiled tori can be
parametrized in a natural way that allows to count the number of square-tiled
tori tiled by a given number of square tiles. There is a natural
$\mathrm{SL}(2,\mathbf{Z})$-action on square-tiled tori and we classify
$\mathrm{SL}(2,\mathbf{Z})$-orbits using two numerical invariants that can be
easily computed. We deduce the exact size of every
$\mathrm{SL}(2,\mathbf{Z})$-orbit. In particular, this answers a question by M.
Bolognesi on the number of cyclic covers of the torus, which corresponds to
particular $\mathrm{SL}(2,\mathbf{Z})$-orbits of square-tiled tori. We also
give the asymptotic behavior of the number of cyclic square-tiled tori.
