A set of bases of a d dimensional complex vector space, each pair of which is unbiased, is a set of mutually unbiased bases (MUBs). MUBs have applications in quantum physics and quantum information theory. Although the motivation to study MUBs comes from physical properties, MUBs are a mathematical structure. This is a mathematical investigation. There are many open problems in the theory of MUBS, some with conjectured solutions. For example: What is the maximum number of MUBs in a d dimensional vector space? Do complete sets of MUBs exist in all dimensions? One such conjectured solution states that a complete set of MUBs exists in a d dimensional complex vector space if and only if a complete set of mutually orthogonal Latin squares (MOLS) of order d exists (Saniga et. al., Journal of Optics B, 6: L19-20, 2004). The aim of this research was to find evidence for or against this conjecture. Inspired by constructions of MUBs that use sets of MOLS, complete sets of MOLS were constructed from two complete sets of MUBs. It is interesting to note that the MOLS structure emerges not from the vectors, but from the inner products of the vectors. Analogous properties between Hjelmslev planes and MUBs, and gaps in this knowledge motivated investigation of Hjelmslev planes. The substructures of a Hjelmslev plane over a Galois ring, and a combinatorial algorithm for generating Hjelmslev planes were developed. It was shown that the analogous properties of Hjelmslev planes and MUBs occur only for odd prime powers, making a strong connection between MUBs and Hjelmslev planes unlikely. A construction of MUBs that uses planar functions was generalised by using an automorphism on the additive group of a Galois field. It is still unclear whether this generalisation is equivalent to the original construction. Relation algebras were constructed from the structure of MUBs which do not share any similarities with relation algebras constructed from MOLS. It is possible that further investigation may yield relation algebras that are similar. It was shown that a set of Wooters and Fields type MUBs, when represented as elements of a group ring, forms a commutative monoid, whereas a set of Alltop type MUBs when similarly represented does not form a closed algebraic structure. It is known that WF and Alltop MUBs are equivalent, thus the lack of a closed structure in the Alltop MUBs suggests that the monoid is not a property of MUBs in general. Complete sets of MOLS and complete sets of MUBs are `similar in spirit', but perhaps this is not an inherent feature of MUBs and MOLS. Since all the known constructions of MUBs rely on algebraic structures which exist only in prime power dimensions, the connection may not be with MOLS, but with algebraic structures which generate both MOLS and MUBs
