  Many definitions of weak n-category have been proposed. It has been widely
observed that each of these definitions is of one of two types: algebraic
definitions, in which composites and coherence cells are explicitly specified,
and non-algebraic definitions, in which a coherent choice of composites and
constraint cells is merely required to exist. Relatively few comparisons have
been made between definitions, and most of those that have concern the
relationship between definitions of just one type. The aim of this thesis is to
establish more comparisons, including a comparison between an algebraic
definition and a non-algebraic definition.
  The thesis is divided into two parts. Part 1 concerns the relationships
between three algebraic definitions of weak n-category: those of Penon and
Batanin, and Leinster's variant of Batanin's definition. A correspondence
between the structures used to define composition and coherence in the
definitions of Batanin and Leinster has long been suspected; we make this
precise for the first time. We use this correspondence to prove several
coherence theorems that apply to all three definitions, and to take the first
steps towards describing the relationship between the weak n-categories of
Batanin and Leinster.
  In Part 2 we take the first step towards a comparison between Penon's
definition of weak n-category and a non-algebraic definition, Simpson's variant
of Tamsamani's definition, in the form of a nerve construction. As a prototype
for this nerve construction, we recall a nerve construction for bicategories
proposed by Leinster, and prove that the nerve of a bicategory given by this
construction is a Tamsamani--Simpson weak 2-category. We then define our nerve
functor for Penon weak n-categories. We prove that the nerve of a Penon weak
2-category is a Tamsamani--Simpson weak 2-category, and conjecture that this
result holds for higher n.
