Shear-free, general-relativistic perfect fluids are investigated in the case where they are either homogeneous or hypersurface-homogeneous (and, in particular, spatially homogeneous). It is assumed that the energy density μ and the presurep of the fluid are related by a barotropic equation of statep = p(μ), where μ +p ≠ 0. Under such circumstances, it follows that either the fluid's volume expansion rate θ or the fluid's vorticity (i.e., rotation) ω must vanish. In the homogeneous case, this leads to only two possibilities: either ω = θ = 0 (the Einstein static solution), or ω ≠ 0,θ = 0 (the Gödel solution). In the hypersurface-homogeneous case, the situation is more complicated: either ω = 0, θ≠ 0 (as exemplified,inter alia, by the Friedmann-Robertson-Walker models), or ω ≠ 0, θ = 0 (which pertains, for example, in general stationary cylindrically symmetric fluids with rigid rotation, or ω = θ = 0 (as occurs for static spherically symmetric solutions). Each possibility is further subdivided in an invariant way, and related to the studies of other authors, thereby unifying and extending these earlier works.