This paper reviews both direct and indirect combustion noise. For convenience, they will simply be referred to as combustion noise and entropy noise. Combustion noise has been studied for well over half a century. However, because of the large number of parameters involved and the complexities inherent in the combustion processes, a widely accepted theory has yet to be developed. For this reason, this review focuses primarily on experimental measurements, semi-empirical relations and empirical but practical prediction methodologies. Important characteristic features and empirical correlations of combustion noise based on open flames and engine noise data are highlighted. Plausible generation of entropy noise by the passage of entropy waves through a nonuniform mean flow was first predicted theoretically circa 1970s. But it took forty years for its existence to be confirmed experimentally. Since then, there have been numerous publications on this subject. They are the primary materials of this review. The fundamental experiment and noise generation mechanism will be discussed first. Of great practical importance is whether there is significant generation of entropy waves inside an engine. This issue and new methods for modeling and predicting internally generated engine entropy noise are items that are examined at some length. Recent advances in computational methods especially in large eddy simulation lead us to envision an important role to be played by high-fidelity numerical simulation in future combustion and entropy noise research and prediction. A critical evaluation of a strategy for investigating and predicting the generation and propagation of these noise components from the combustor of an engine to its exhaust is included in this review.