  Truncated abstract: The formation of a protostellar disc is a natural outcome
during the star formation process. As gas in a molecular cloud core collapses
under self-gravity, the angular momentum of the gas will slow its collapse on
small scales and promote the formation of a protostellar disc. Although the
angular momenta of dense star-forming cores remain to be fully characterized
observationally, existing data indicates that typical cores have enough angular
momenta to form relatively large, rotationally supported discs. However,
molecular clouds are observed to be permeated by magnetic fields, which can
strongly affect the evolution of angular momentum through magnetic braking.
Indeed, in the ideal MHD limit, magnetic braking has been shown to be so
efficient as to remove essentially all of the angular momentum of the material
close to the forming star such that disc formation is suppressed. This is known
as the magnetic braking catastrophe. The catastrophe must be averted in order
for the all-important rotationally supported discs to appear, but when and how
this happens remains debated. We review the resolutions proposed to date, with
emphasis on misalignment, turbulence and especially non-ideal effects. The
dissipative non-ideal effects weaken the magnetic field, and the dispersive
term redirects it to promote or hinder disc formation. When self-consistently
applying non-ideal processes, rotationally supported discs of at least tens of
au form, thus preventing the magnetic braking catastrophe. The non-ideal
processes are sensitive to the magnetic field strength, cosmic ray ionization
rate, and gas and dust grain properties, thus a complete understanding of the
host molecular cloud is required. Therefore, the properties of the host
molecular cloud -- and especially its magnetic field -- cannot be ignored when
numerically modelling the formation and evolution of protostellar discs.
