  This thesis concerns the study of homogeneous factorisations of complete
graphs with edge-transitive factors. A factorisation of a complete graph $K_n$
is a partition of its edges into disjoint classes. Each class of edges in a
factorisation of $K_n$ corresponds to a spanning subgraph called a factor. If
all the factors are isomorphic to one another, then a factorisation of $K_n$ is
called an isomorphic factorisation. A homogeneous factorisation of a complete
graph is an isomorphic factorisation where there exists a group $G$ which
permutes the factors transitively, and a normal subgroup $M$ of $G$ such that
each factor is $M$-vertex-transitive. If $M$ also acts edge-transitively on
each factor, then a homogeneous factorisation of $K_n$ is called an
edge-transitive homogeneous factorisation. The aim of this thesis is to study
edge-transitive homogeneous factorisations of $K_n$. We achieve a nearly
complete explicit classification except for the case where $G$ is an affine
2-homogeneous group of the form $Z_p^R \rtimes G_0$, where $G_0 \leq \Gamma
L(1,p^R)$. In this case, we obtain necessary and sufficient arithmetic
conditions on certain parameters for such factorisations to exist, and give a
generic construction that specifies the homogeneous factorisation completely,
given that the conditions on the parameters hold. Moreover, we give two
constructions of infinite families of examples where we specify the parameters
explicitly. In the second infinite family, the arc-transitive factors are
generalisations of certain arc-transitive, self-complementary graphs
constructed by Peisert in 2001.
