Shear-free normal cosmological models are the perfect fluid solutions of Einstein’s equations in which rotation and shear vanish, and which are not static [they were all found by A. Barnes, Gen. Relativ. Gravit. 4, 105 (1973)]. They are either spherically, plane, or hyperbolically symmetric. Their symmetries are discussed in various coordinate systems and related to the conformal group of the three-dimensional flat space. A coordinate representation is introduced which unites all three cases into a single two-parameter family. The limiting transitions to the Friedman–Lemaitre–Robertson–Walker (FLRW) models and to the Schwarzschild–de Sitter-like solutions are presented.