The Burgers equation is the simplest nonlinear generalization of the diffusion equation. We present a detailed dynamical renormalization-group analysis of this equation subject to random noise. The noise itself can be the product of another stochastic process and is hence allowed to have correlations in space and/or time. In dimensions higher than a critical dc weak and strong noise lead to different scaling exponents, while for d<dc any amount of noise is relevant resulting in strong-coupling behavior. In the absence of temporal correlations we find two regimes for d<dc: either the hydrodynamic behavior is determined by white noise and correlations are unimportant, or correlations dominate and the resulting scaling exponents can be obtained exactly. With temporal correlations present, the hydrodynamic behavior is much more complex, as renormalization predicts a complicated dependence of the effective noise spectrum on frequency in certain regimes. The relevance of these results to two interesting problems is discussed. One is the anomalous transverse fluctuations of a directed polymer in a random medium, and the other is a description of a growing interface. Various recent numerical simulations are reviewed in the light of these results. For example, we show that an exponent identity observed in all simulations so far follows simply from the Galilean invariance of the equation in the absence of temporal correlations.