  When the universe is treated as a quantum system, it is described by a wave
function. This wave function is a function not only of the matter fields, but
also of spacetime. The no-boundary proposal is the idea that the wave function
should be calculated by summing over geometries that have no boundary to the
past, and over regular matter configurations on these geometries. Accordingly,
the universe is finite, self-contained and the big bang singularity is avoided.
Moreover, given a dynamical theory, the no-boundary proposal provides
probabilities for various solutions of the theory. In this sense it provides a
quantum theory of initial conditions.
  This review starts with a general overview of the framework of quantum
cosmology, describing both the canonical and path integral approaches, and
their interpretations. After recalling several heuristic motivations for the
no-boundary proposal, its consequences are illustrated with simple examples,
mainly in the context of cosmic inflation. We review how to include
perturbations, assess the classicality of spacetime and how probabilities may
be derived. A special emphasis is given to explicit implementations in
minisuperspace, to observational consequences, and to the relationship of the
no-boundary wave function with string theory. At each stage, the required
analytic and numerical techniques are explained in detail, including the
Picard-Lefschetz approach to oscillating integrals.
