The development and integration in the electrical grid of new types of energy (solar, wind, marine, biomass and geothermal) create power fluctuations due to their intermittent character of operation. Considering the high demand of renewable energies, hydro power plants are working with an increasing number of start and stop cycles to balance the electric network. The need for a short/long-term operating strategy for hydro power plants (in terms of operation but especially in terms of costs) has led to the emergence of different models in the hydro power industry to estimate the cost incurred by start and stop cycles. This paper presents a critical literature review of the different existing approaches for estimating the start and stop costs of hydro power plants. Few articles and technical reports can be found in the open literature due to the complexity of the subject and its relative novelty in the context of the hydro power industry. We propose to categorize the existing approaches in two types: economic and technicaleconomic. The economic approach propose an estimation of cost without investigating in details the degradation phenomena of the equipment of a hydro power plant caused by start and stop cycles. On the contrary, the technicaleconomic approach aim to estimate the degradation caused by start and stop cycles events, using expert judgement and database analysis. The link with the cost is made by assuming a share of the total cost of maintenance and operation for start and stop or it is established by quantifying an equivalence between a start and stop event and a number of normal operating hours in terms of damage incurred to the equipment. The final paper discusses and compares these approaches, giving an overview and opening research perspectives on a serious and current issue that hydro power plants owners and designers face nowadays: the continuously increasing number of start and stops and how much it costs.