  The dynamics of quantum systems far from equilibrium represents one of the
most challenging problems in theoretical many-body physics. While the evolution
is in general intractable in all its details, relevant observables can become
insensitive to microscopic system parameters and initial conditions. This is
the basis of the phenomenon of universality. Far from equilibrium, universality
is identified through the scaling of the spatiotemporal evolution of the
system, captured by universal exponents and functions. Theoretically, this has
been studied in examples as different as the reheating process in inflationary
universe cosmology, the dynamics of nuclear collision experiments described by
quantum chromodynamics, or the post-quench dynamics in dilute quantum gases in
non-relativistic quantum field theory. Here we observe the emergence of
universal dynamics by evaluating spatially resolved spin correlations in a
quasi one-dimensional spinor Bose-Einstein condensate. For long evolution times
we extract the scaling properties from the spatial correlations of the spin
excitations. From this we find the dynamics to be governed by transport of an
emergent conserved quantity towards low momentum scales. Our results establish
an important class of non-stationary systems whose dynamics is encoded in time
independent scaling exponents and functions signaling the existence of
non-thermal fixed points. We confirm that the non-thermal scaling phenomenon
involves no fine-tuning, by preparing different initial conditions and
observing the same scaling behaviour. Our analog quantum simulation approach
provides the basis to reveal the underlying mechanisms and characteristics of
non-thermal universality classes. One may use this universality to learn, from
experiments with ultra-cold gases, about fundamental aspects of dynamics
studied in cosmology and quantum chromodynamics.
