  In this paper we study pattern avoidance in Latin Squares, which gives us a
two dimensional analogue of the well studied notion of pattern avoidance in
permutations. Our main results include enumerating and characterizing the Latin
Squares which avoid patterns of length three and a generalization of the
Erd\H{o}s-Szekeres theorem. We also discuss equivalence classes among longer
patterns, and conclude by describing open questions of interest both in light
of pattern avoidance and their potential to reveal information about the
structure of Latin Squares. Along the way, we show that classical results need
not trivially generalize, and demonstrate techniques that may help answer
future questions.
