  This thesis is dedicated to studying cosmological inflation, which is a
period of accelerated expansion in the very early Universe that is required to
explain the observed anisotropies in the cosmic microwave background.
Inflation, when combined with quantum mechanics, also provides the
over-densities that grow into the structure of the modern Universe.
Understanding perturbations during this period of inflation is important, and
we study these perturbations in detail in this work. We will assume that
inflation is driven by a single scalar field, called the inflaton. When the
shape of the potential energy is flat, the inflaton can enter a phase of
"ultra-slow-roll inflation". We study the stability of such a period of
inflation, and find that it can be stable and long-lived, although it has a
dependence on the initial velocity of the inflaton field. This is different to
the slow-roll regime of inflation, which is always stable, but has no
dependence on the initial velocity. In the second part of this thesis, we use
the stochastic formalism for inflation in order to take account of the
non-perturbative backreaction of quantum fluctuations during inflation. We use
this formalism to study curvature fluctuations during inflation, and we derive
full probability distributions of these fluctuations. This allows us to study
the statistics of large fluctuations that can lead to the formation of rare
objects, such as primordial black holes. In general, we find that when the
quantum effects modelled by the stochastic formalism are correctly accounted
for, many more primordial black holes can be formed than one would expect if
these quantum effects were not taken into account.
