This thesis consists of the four papers listed below and a survey of the research area.

I Lina J. Andrén: Avoiding (m, m, m)-arrays of order n = 2^{k}

II Lina J. Andrén: Avoidability of random arrays

III Lina J. Andr´en: Avoidability by Latin squares of arrays with even order

IV Lina J. Andrén, Carl Johan Casselgren and Lars-Daniel Öhman: Avoiding arrays of odd order by Latin squares

Papers I, III and IV are all concerned with a conjecture by Häggkvist saying that there is a constant c such that for any positive integer n, if m ≤ cn, then for every n × n array A of subsets of {1, . . . , n} such that no cell contains a set of size greater than m, and none of the elements 1, . . . , n belongs to more than m of the sets in any row or any column of A, there is a Latin square L on the symbols 1, . . . , n such that there is no cell in L that contains a symbol that belongs to the set in the corresponding cell of A. Such a Latin square is said to avoid A. In Paper I, the conjecture is proved in the special case of order n = 2^{k} . Paper III improves on the techniques of Paper I, expanding the proof to cover all arrays of even order. Finally, in Paper IV, similar methods are used together with a recoloring theorem to prove the conjecture for all orders. Paper II considers another aspect of the problem by asking to what extent way a deterministic result concerning the existence of Latin squares that avoid certain arrays can be used when the sets in the array are assigned randomly.