Revent developments in quantum cosmology

In quantum cosmology, the functional psi which is the probability amplitude of finding a given field configuration and a given three-metric on a compact spacelike three-surface S obeys a zero energy Schrodinger equation, the Wheeler-De Witt equation for closed universes. psi may be given by a functional integral taken over all compact euclidean four-metrics which have a boundary at S, and over all field configurations which match the given value on S. After a re-examination of the Wheeler-De Witt equation for a closed FRW universe in which there is a massive scalar field phi minimally coupled to the gravitational field, I calculate the time-evolution of the cosmic scale factor and of the scalar field in the case of a lambda * phi^{n} theory, in order to understand when a lambda * phi^{n} theory is able to drive inflation. Some new topics are mentioned, too.