  We develop a limit theory of Latin squares, paralleling the recent limit
theories of dense graphs and permutations. We introduce a notion of density, an
appropriate version of the cut distance, and a space of limit objects -
so-called Latinons. Key results of our theory are the compactness of the limit
space and the equivalence of the topologies induced by the cut distance and the
left-convergence. Last, using Keevash's recent results on combinatorial
designs, we prove that each Latinon can be approximated by a finite Latin
square.
