  We study the cosmological effects of adding terms of higher-order in the
usual energy-momentum tensor to the matter lagrangian of general relativity.
This is in contrast to most studies of higher-order gravity which focus on
generalising the Einstein-Hilbert curvature contribution to the lagrangian. The
resulting cosmological theories include many particular theories, like bulk
viscous cosmologies, loop quantum gravity, K-essence, and brane-world
cosmologies. We find a range of exact solutions for isotropic universes,
discuss their behaviours with reference to the early and late-time evolution,
accelerated expansion, and the occurrence or avoidance of singularities. We
briefly discuss extensions to anisotropic cosmologies and delineate the
situations where the higher-order matter terms will dominate over anisotropies
on approach to cosmological singularities.
