We review uncertainty quantification (UQ) for hyperbolic systems of conservation (balance) laws. The input uncertainty could be in the initial data, fluxes, coefficients, source terms or boundary conditions. We focus on forward UQ or uncertainty propagation and review deterministic methods such as stochastic Galerkin and stochastic collocation finite volume methods for approximating random (field) entropy solutions. Statistical sampling methods of the Monte Carlo and multilevel Monte Carlo (MLMC) type are also described. We present alternative UQ frameworks such as measure-valued solutions and statistical solutions.