  Universal algebra uniformly captures various algebraic structures, by
expressing them as equational theories or abstract clones. The ubiquity of
algebraic structures in mathematics and related fields has given rise to
several variants of universal algebra, such as symmetric operads, non-symmetric
operads, generalised operads, and monads. These variants of universal algebra
are called notions of algebraic theory. In the first part of this thesis, we
develop a unified framework for notions of algebraic theory which includes all
of the above examples. Our key observation is that each notion of algebraic
theory can be identified with a monoidal category, in such a way that theories
correspond to monoid objects therein. We introduce a categorical structure
called metamodel, which underlies the definition of models of theories. We also
consider morphisms between notions of algebraic theory, which are a monoidal
version of profunctors. Every strong monoidal functor gives rise to an adjoint
pair of such morphisms, and provides a uniform way to establish isomorphisms
between categories of models in different notions of algebraic theory. A
general structure-semantics adjointness result and a double categorical
universal property of categories of models are also shown.
  In the second part of this thesis, we shift from the general study of
algebraic structures, and focus on a particular algebraic structure: higher
dimensional categories. Among several existing definitions of higher
dimensional categories, we choose to look at the one proposed by Batanin and
later refined by Leinster. We show that the notion of extensive category plays
a central role in Batanin and Leinster's definition. Using this, we generalise
their definition by allowing enrichment over any locally presentable extensive
category.

Category Theory