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The row-column setting for comparing v experimental treatments is a p × q array of experimental units, with the p rows and q columns representing levels of two blocking factors. When p = q = v, the optimal design for estimating treatment contrasts is a Latin square. For a v ^t factorial treatment set in the same setting, t mutually orthogonal Latin squares are an optimal main-effects plan. Orthogonal collections of Latin squares arise when these results are generalized to b row-column layouts of size v × v, where b is greater than 1. This article studies these orthogonal collections, with special emphasis on small v and b.