  In this thesis we present research into linear perturbations in
Lemaitre-Tolman-Bondi (LTB) and Assisted Coupled Quintessence (ACQ)
Cosmologies. First we give a brief overview of the standard model of cosmology.
We then introduce Cosmological Perturbation Theory (CPT) at linear order for a
flat Friedmann-Robertson-Walker (FRW) cosmology. Next we study linear
perturbations to a Lemaitre-Tolman-Bondi (LTB) background spacetime. Studying
the transformation behaviour of the perturbations under gauge transformations,
we construct gauge invariant quantities in LTB. We show, using the perturbed
energy conservation equation, that there is a conserved quantity in LTB which
is conserved on all scales. We then briefly extend our discussion to the
Lemaitre spacetime, and construct gauge-invariant perturbations in this
extension of LTB spacetime. We also study the behaviour of linear perturbations
in assisted coupled quintessence models in a FRW background. We provide the
full set of governing equations for this class of models, and solve the system
numerically. The code written for this purpose is then used to evolve growth
functions for various models and parameter values, and we compare these both to
the standard $\Lambda$CDM model and to current and future observational bounds.
We also examine the applicability of the "small scale approximation", often
used to calculate growth functions in quintessence models, in light of upcoming
experiments such as SKA and Euclid. We find the results of the full equations
deviates from the approximation by more than the experimental uncertainty for
these future surveys. The construction of the numerical code, PYESSENCE,
written in Python to solve the system of background and perturbed evolution
equations for assisted coupled quintessence, is also discussed.
