Reducing primary energy consumption by utilizing heat recovery systems has become increasingly important in industry. In many sectors, heating and cooling is required at different temperature levels at the same time. For this purpose, heat pumps are highly attractive energy conversion devices. Heat pumps are widely used for refrigeration, air-conditioning, space heating, hot water production, heat upgrading, or waste heat recovery. The aim of this paper is to review the literature for mechanical driven heat pumps and refrigeration systems with focus on multi-temperature applications. Different design strategies are presented, including cycles with multi-stage compressors, (multiple) ejectors, expansion valves, cascades (with secondary loops), and separated gas coolers. This review highlights the major advantages, challenges, and industrial applications of each multi-temperature heat pump cycle family. Schematics and pressure-enthalpy diagrams illustrate the most promising cycles. The performance of the cycles is compared in terms of First Law efficiency (COP) and Second Law efficiency (exergy) using simplified thermodynamic simulations. The literature reveals that the major part (approximately 70%) of multi-temperature heat pump applications are found in refrigeration, i.e. supermarket food cooling, household fridges/freezers, and cooling/air-conditioning/storage during transportation. In contrast, studies on multi-temperature heating applications are rather rare with the exception of space floor heating and hot water production. Most multi-temperature cycle designs use two heat sources or two heat sinks. Heat pumps with more than three stages are not common, except for natural gas liquefaction. In supermarket applications, multiple compressors with transcritical CO2 are an established key technology. Cascades with secondary loops are another frequently applied system, mostly in the USA. Cycles with multiple ejectors are ready to market and seem to be a promising modification for system performance improvement. Ejector cycles in refrigeration and air-conditioning systems are still under development. Expansion valve cycles are an established technology in household refrigeration. Separated gas coolers for space and hot water heating have recently attracted attention due to the possible combination with supercritical CO2 cycles. Overall, this review paper serves to select the most appropriate multi-temperature heat pump cycle for a specific application.