A comprehensive analysis of gened relativistic spacetimes which admit a shearfree, irrotational and geodesic time-like coogruence is presented. The equations governing the models for a gened energy-momentum tensor are wrilten down. Coordinates in which the metric of such spacetimes takes on a simplified form are established. The general subwses of 'zero anisotropic stress', 'zero heat-flux vector' and 'two-component Ruids' 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. Anisowopic fluid models and viscous fluid models with heat conduction are examined. Also. models with B perfect Ruid plus a magnetic field or with pure radiation. and models with two non-collinw perfect fluids (satisfying a variety of physical conditions) are inwtigated. In particular. models with a (single) perfect Ruid which is tilting with respect to the shem-free, vorticity-free and acceleration-free time-like congruence are discussed.PACS numbers: 0420J. 9880H