The majority of this thesis is an investigation of algebras arising from K-perfect m-cycle systems. The ﬁrst two chapters contain combinatorial results, while the next two give the algebraic results. The ﬁrst chapter contains results on edge-coloured graph decompositions. Using a re¬sult from , we ﬁnd necessary and suﬃcient conditions for the existence of a non-trivial complete edge-coloured graph which has a decomposition into ‘isomorphic’ copies of a given edge-coloured graph. A suﬃcient condition is also given for ensuring the embedding of partial decompositions in complete decompositions of ﬁnite order. In the second chapter, K-perfect m-circuit systems and K-perfect m-cycle systems are introduced. Results from the second chapter show the existence of K-perfect m-cycle sys¬tems, and also of a particular class of K-perfect m-circuit systems, for general K and m.It is also shown that every partial K-perfect m-cycle system embeds in a K-perfect m-cycle system of ﬁnite order. The proofs of these results are based on results in the ﬁrst chapter. The third chapter discusses a type of algebra that arises from a K-perfect m-cycle system. These algebras generalise the Steiner quasigroups which arise from Steiner triple systems (that is 3-cycle systems). It is determined when the class of algebras arising from the K-perfect m-cycle systems is a variety. This generalises the results of  and . The fourth chapter continues investigation of the variety generated by algebras arising from K-perfect m-cycle systems. It is shown that the free algebras of this variety arise from cycle systems. It also shown that the algebras arising from ﬁnite K-perfect m-cycle systems, and those arising from ﬁnite K-perfect m-cycle systems with K� ⊆ K, generate the same variety. Two questions asked in  are also answered. In the ﬁfth chapter we introduce the problem of determining the parameters for which there exist two arrays, each row-Latin and column-Latin, which can be superimposed to obtain every 2-element subset of an n-set exactly once. A complete solution in the case that n is odd is given. There are also partial results for the case of even n, and discussion and results on generalisations of this problem.