This thesis considers problems concerning Latin squares and sets of mutually quasiorthogonal Latin squares (MQOLS). We show how MQOLS are related to a number of other designs and establish bounds on Nq(n), the maximum number of Latin squares of order n in a mutually quasi-orthogonal set. We report the number of quasi-complete mappings admitted by each group of order 15 or less, and explain the surprising result that each of the non-cyclic groups of order 8 possesses exactly 384 complete mappings. For each group G of order 15 or less, we identify the sizes of all maximal sets of mutually orthogonal orthomorphisms. We also identify a number of new maximal sets for larger groups. We present a method to determine all proper, maximal sets of MQOLS of order n and carry this out for n < 6. Also we present a search for 3 MQOLS of order 10, which, whilst not identifying such a set, led to the identification of all resolutions of each (10,3, 2)-balanced incomplete block design. We give a construction for MQOLS based on groups, and use this to determine new sets of 2n - 1 MQOLS of order 2n based on two infinite classes of group. Existence results for MQOLS based on groups are also extended. Two constructions for (n x n)/k semi-Latin squares are given, one of which provides some new A-, D- and E-optimal examples with k > n which out-perform the existing A-, D- and E-optimal examples in the E'-criteria. Finally we consider the problem of determining invertible directed terraces of each non-abelian group of order < 21, and in so doing construct the first doubly balanced bipartite tournament of odd order.