  We investigate the Liouvillian integrability of Hamiltonian systems
describing a universe filled with a scalar field (possibly complex). The tool
used is the differential Galois group approach, as introduced by Morales-Ruiz
and Ramis. The main result is that the generic systems are non-integrable,
although there still exist some values of parameters for which integrability
remains undecided. We also draw a connection with chaos present in such
cosmological models. The first part of the article deals with minimally coupled
fields, and the second treats the conformal couping.
