For fluctuating thermodynamic currents in non-equilibrium steady states, the thermodynamic uncertainty relation expresses a fundamental trade-off between precision, i.e. small fluctuations, and dissipation. Using large deviation theory, we show that this relation follows from a universal bound on current fluctuations that is valid beyond the Gaussian regime and in which only the total rate of entropy production enters. Variants and refinements of this bound hold for fluctuations on finite time scales and for Markovian networks with known topology and cycle affinities. Applied to molecular motors and heat engines, the bound on current fluctuations imposes constraints on the efficiency and power.