These notes aim to provide a pedagogical introduction to the subject of quantum cosmology: the study of the very early universe including quantum effects. We assume previous knowledge of general relativity and quantum field theory. Basic knowledge in cosmology is ideal but for those unfamiliar with this topic we briefly review the key notions in the introduction (good lecture notes and books more comprehensive on this topic are for instance Baumann 2009 and Mukhanov 2005).

We start by reviewing the early attempt to canonically quantize gravity. We explain how this led to the concept of a wavefunction of the universe together with the Wheeler-de Witt equation governing it. Then we turn to the no-boundary and tunneling proposals and review the many successive endeavors to make these proposals concrete in simplified models. We finally study in detail the Euclidean and Lorentzian gravitational path integrals, including some tools used in this context such as the BRST procedure and the Picard-Lefschetz theory of complex analysis.