  In this dissertation, we study the evolution and properties of the relic (or
cosmic) neutrino distribution from neutrino freeze-out at $T=O(1)$ MeV through
the free-streaming era up to today, focusing on the deviation of the neutrino
spectrum from equilibrium. In particular, we demonstrate the presence of
chemical non-equilibrium that continues to the present day. The work naturally
separates into two parts. The first focuses on aspects of the relic neutrinos
that can be explored using conservation laws. The second part studies the
neutrino distribution using the full general relativistic Boltzmann equation.
  Part one begins with an overview of the history of the Universe, from just
prior to neutrino freeze-out up through the present day, placing the history of
cosmic neutrino evolution in its proper context. Motivated by the Planck CMB
measurements of the effective number of neutrinos, we derive those properties
of neutrino freeze-out that depend only on conservation laws and are
independent of the details of the scattering processes. Part one ends with a
characterization of the present day neutrino spectrum as seen from Earth.
  The second part of this dissertation focuses on the properties of cosmic
neutrinos that depend on the details of the neutrino reactions, as is necessary
for modeling the non-thermal distortions from equilibrium and computing
freeze-out temperatures. We detail a new spectral method for solving the
Boltzmann equation, based on a dynamical basis of orthogonal polynomials, as
well as an improved procedure for analytically simplifying the corresponding
scattering integrals for subsequent numerical computation. We apply these novel
solution methods to solve the Boltzmann equation through the neutrino
freeze-out period and perform parametric studies of the dependence of neutrino
freeze-out on standard model parameters.
