  Cosmological perturbation theory is crucial for our understanding of the
universe. The linear theory has been well understood for some time, however
developing and applying the theory beyond linear order is currently at the
forefront of research in theoretical cosmology. This thesis studies the
applications of perturbation theory to cosmology and, specifically, to the
early universe. Starting with some background material introducing the
well-tested 'standard model' of cosmology, we move on to develop the formalism
for perturbation theory up to second order giving evolution equations for all
types of scalar, vector and tensor perturbations, both in gauge dependent and
gauge invariant form. We then move on to the main result of the thesis, showing
that, at second order in perturbation theory, vorticity is sourced by a
coupling term quadratic in energy density and entropy perturbations. This
source term implies a qualitative difference to linear order. Thus, while at
linear order vorticity decays with the expansion of the universe, the same is
not true at higher orders. This will have important implications on future
measurements of the polarisation of the Cosmic Microwave Background, and could
give rise to the generation of a primordial seed magnetic field. Having derived
this qualitative result, we then estimate the scale dependence and magnitude of
the vorticity power spectrum, finding, for simple power law inputs a small,
blue spectrum. The final part of this thesis concerns higher order perturbation
theory, deriving, for the first time, the metric tensor, gauge transformation
rules and governing equations for fully general third order perturbations. We
close with a discussion of natural extensions to this work and other possible
ideas for off-shooting projects in this continually growing field.
