Reservoir operation optimization aims to determine release and transfer decisions that maximise water management objectives such as ensuring a reliable water supply, hydropower production, mitigation of downstream floods, etc. An extensive and growing body of scientific literature exists on advancing and applying mathematical optimization methods to reservoir operation problems. In this paper, we review such literature according to a novel classification system of optimization approaches, which focuses on the characteristics of the actual operation problem – i.e. what needs to be optimized, or in mathematical terms, ‘the argument’ of the optimization problems - rather than the mathematical properties of the optimization algorithm. This enables us to discuss the advantages, limitations and the scope of application of the different optimisation methods; and to provide practical guidelines for matching the properties of a system and operation problem with a suitable optimization method. Alongside this paper we provide code to implement many of the methods we review for an illustrative reservoir system.