The collective behaviour of a many-body system near a continuous phase transition is insensitive to the details of its microscopic physics; for example, thermodynamic observables follow generalized scaling laws near the phase transition(1). The Berezinskii-Kosterlitz-Thouless (BKT) phase transition(2,3) in two-dimensional Bose gases presents a particularly interesting case because the marginal dimensionality and intrinsic scaling symmetry(4) result in a broad fluctuation regime and an extended range of universal scaling behaviour. Studies of the BKT transition in cold atoms have stimulated great interest in recent years(5-10), but a clear demonstration of critical behaviour near the phase transition has remained elusive. Here we report in situ density and density-fluctuation measurements of two-dimensional Bose gases of caesium at different temperatures and interaction strengths, observing scale-invariant, universal behaviours. The extracted thermodynamic functions confirm the existence of a wide universal region near the BKT phase transition, and provide a sensitive test of the universality predicted by classical-field theory(11,12) and quantum Monte Carlo calculations(13). Our experimental results provide evidence for growing density-density correlations in the fluctuation region, and call for further explorations of universal phenomena in classical and quantum critical physics.