The extraction of bioactive compounds of pharmaceutical interest from natural sources has been significantly explored in recent decades. However, the extraction techniques used were not very efficient in terms of time and energy consumption; additionally, the solvents used for the extraction were harmful for the environment. To improve the environmental impact of the extractions and at the same time increase the extraction yields, several new extraction techniques were developed. Among the most used ones are ultrasound-assisted extraction and microwave-assisted extraction. These extraction techniques increased the yield and selectivity of the extraction in a smaller amount of time with a decrease in energy consumption. Nevertheless, a high volume of organic solvents was still used for the extraction, causing a subsequent environmental problem. Neoteric solvents appeared as green alternatives to organic solvents. Among the neoteric solvents, deep eutectic solvents were evidenced to be one of the best alternatives to organic solvents due to their intrinsic characteristics. These solvents are considered green solvents because they are made up of natural compounds such as sugars, amino acids, and carboxylic acids having low toxicity and high degradability. In addition, they are simple to prepare, with an atomic economy of 100%, with attractive physicochemical properties. Furthermore, the huge number of compounds that can be used to synthesize these solvents make them very useful in the extraction of bioactive compounds since they can be tailored to be selective towards a specific component or class of components. The main aim of this paper is to give a comprehensive review which describes the main properties, characteristics, and production methods of deep eutectic solvents as well as its application to extract from natural sources bioactive compounds with pharmaceutical interest. Additionally, an overview of the more recent and sustainable extraction techniques is also given.