  We review the canonical quantisation of the geometry of the spacetime in the
cases of a simply and a non-simply connected manifold. In the former, we
analyse the information contained in the solutions of the Wheeler-DeWitt
equation and interpret them in terms of the customary boundary conditions that
are typically imposed on the semiclassical wave functions. In particular, we
review three different paradigms for the quantum creation of a homogeneous and
isotropic universe. For the quantisation of a non-simply connected manifold the
best framework is the so-called third quantisation formalism, in which the wave
function of the universe is seen as a field that propagates in the space of
Riemannian $3$-geometries, which turns out to be isomorphic to a (part of a)
$1+5$ Minkowski spacetime. Thus, the quantisation of the wave function follows
the customary formalism of a quantum field theory. A general review of the
formalism is given and it is analysed the creation of the universes, their
initial expansion and the appearance of matter after inflation. These features
are presented in more detail in the case of a homogeneous and isotropic
universe. The main conclusion in both cases is that the most natural way in
which the universes should be created is in entangled universe-antiuniverse
pairs.
