A comprehensive analysis of general relativistic spacetimes which admit a shear-free, irrotational and geodesic time-like congruence is presented. The equations governing the models for a general energy--momentum tensor are written down. Coordinates in which the metric of such spacetimes takes on a simplified form are established. The general subcases of `zero anisotropic stress', `zero heat-flux vector' and `two-component fluids' are investigated. In particular, perfect-fluid Friedmann--Robertson--Walker models and spatially homogeneous models are discussed. Models with a variety of physically relevant energy--momentum tensors are considered. Anisotropic fluid models and viscous fluid models with heat conduction are examined. Also, models with a perfect fluid plus a magnetic field or with pure radiation, and models with two non-collinear perfect fluids (satisfying a variety of physical conditions) are investigated. In particular, models with a (single) perfect fluid which is tilting with respect to the shear-free, vorticity-free and acceleration-free time-like congruence are discussed.