  For current fluctuations in non-equilibrium steady states of Markovian
processes, we derive four different universal bounds valid beyond the Gaussian
regime. Different variants of these bounds apply to either the entropy change
or any individual current, e.g., the rate of substrate consumption in a
chemical reaction or the electron current in an electronic device. The bounds
vary with respect to their degree of universality and tightness. A universal
parabolic bound on the generating function of an arbitrary current depends
solely on the average entropy production. A second, stronger bound requires
knowledge both of the thermodynamic forces that drive the system and of the
topology of the network of states. These two bounds are conjectures based on
extensive numerics. An exponential bound that depends only on the average
entropy production and the average number of transitions per time is rigorously
proved. This bound has no obvious relation to the parabolic bound but it is
typically tighter further away from equilibrium. An asymptotic bound that
depends on the specific transition rates and becomes tight for large
fluctuations is also derived. This bound allows for the prediction of the
asymptotic growth of the generating function. Even though our results are
restricted to networks with a finite number of states, we show that the
parabolic bound is also valid for three paradigmatic examples of driven
diffusive systems for which the generating function can be calculated using the
additivity principle. Our bounds provide a new general class of constraints for
nonequilibrium systems.
