Understanding the behaviour of isolated quantum systems far from equilibrium and their equilibration is one of the most pressing problems in quantum many-body physics(1,2). There is strong theoretical evidence that sufficiently far from equilibrium a wide variety of systems-including the early Universe after inflation(3-6) quark-gluon matter generated in heavy-ion collisions(7-9), and cold quantum gases(4 ,10-14)- exhibit universal scaling in time and space during their evolution, independent of their initial state or microscale properties. However, direct experimental evidence is lacking. Here we demonstrate universal scaling in the time-evolving momentum distribution of an isolated, far-from-equilibrium, onedimensional Bose gas, which emerges from a three-dimensional ultracold Bose gas by means of a strong cooling quench. Within the scaling regime, the time evolution of the system at low momenta is described by a time-independent, universal function and a single scaling exponent. The non-equilibrium scaling describes the transport of an emergent conserved quantity towards low momenta, which eventually leads to the build-up of a quasi-condensate. Our results establish universal scaling dynamics in an isolated quantum many-body system, which is a crucial step towards characterizing time evolution far from equilibrium in terms of universality classes. Universality would open the possibility of using, for example, coldatom set-ups at the lowest energies to simulate important aspects of the dynamics of currently inaccessible systems at the highest energies, such as those encountered in the inflationary early Universe.