  Neutrino physics in the early Universe is key to our understanding of later
cosmological stages, such as primordial nucleosynthesis (BBN) or the formation
of large-scale structures. The coming decade promises new experimental results
to explore and constrain cosmological models even more precisely - which
requires robust theoretical predictions. This PhD thesis presents a study of
the evolution of neutrinos in the first seconds after the Big Bang, more
precisely when the temperature of the Universe is of the order of one
mega-electronvolt. This evolution is obtained numerically by solving kinetic
equations for which we propose a new derivation. A first application is the
calculation of the so-called "standard" decoupling in order to calculate the
cosmological parameter quantifying the energy density of the primordial
relativistic species, $N_\mathrm{eff}$, to a precision of a few
ten-thousandths. This study has highlighted the possibility of effectively
describing the phenomenon of flavour oscillations, taking advantage of the
large separation of time scales involved. Such an approximation is then adapted
and validated in the case of non-zero asymmetries between neutrinos and
antineutrinos. Finally, we study semi-analytically the consequences of
incomplete neutrino decoupling on BBN, in order to understand how the
primordial abundances of helium and deuterium are affected by this physics.
