Climate models are widely used in climate change impact studies. However, these simulations often cannot be used directly due to inherent limitations, such as structural biases or parametric uncertainties. Nevertheless, several so-called "bias correction" (B-C) or "bias adjustment" methods have been proposed to get these simulations closer to real observations. Various studies have reviewed available methods; however, numerous innovative methods have been developed in recent years. An up-to-date review of the B-C methods is presented here. To compare these complex methods, a focus is placed on the pedagogy of the presentation. The main lines of thought are presented based on the method assumptions, mathematical form, properties, and applicative purposes. Six representative quantile-based methods are compared for temperature and precipitation monthly time series over the European area, for a climate change scenario with a strong CO2 forcing which is chosen here to facilitate the analysis of the differences among the methods. New, simple, and easy-to-understand diagnostic tools are recommended to measure the impact of the adjustment on the ability of B-C methods to: (1) bring the model outputs closer to observations over the historical record, (2) exploit as much as possible the climate change signal provided by the model. Each B-C method is intended to find the best compromise between these two objectives. A discussion on potential pathways for future developments is finally proposed.