Over the last years, we observed a significant increase in the number of published studies that focus on the synthesis and characterization of deep eutectic solvents (DESs). These materials are of particular interest mainly due to their physical and chemical stability, low vapor pressure, ease of synthesis, and the possibility of tailoring their properties through dilution or change of the ratio of parent substances (PS). DESs, considered as one of the greenest families of solvents, are used in many fields, such as organic synthesis, (bio)catalysis, electrochemistry, and (bio)medicine. DESs applications have already been reported in various review articles. However, these reports mainly described these components’ basics and general properties without focusing on the particular, PS-wise, group of DESs. Many DESs investigated for potential (bio)medical applications comprise organic acids. However, due to the different aims of the reported studies, many of these substances have not yet been investigated thoroughly, which makes it challenging for the field to move forward. Herein, we propose distinguishing DESs comprising organic acids (OA-DESs) as a specific group derived from natural deep eutectic solvents (NADESs). This review aims to highlight and compare the applications of OA-DESs as antimicrobial agents and drug delivery enhancers—two essential fields in (bio)medical studies where DESs have already been implemented and proven their potential. From the survey of the literature data, it is evident that OA-DESs represent an excellent type of DESs for specific biomedical applications, owing to their negligible cytotoxicity, fulfilling the rules of green chemistry and being generally effective as drug delivery enhancers and antimicrobial agents. The main focus is on the most intriguing examples and (where possible) application-based comparison of particular groups of OA-DESs. This should highlight the importance of OA-DESs and give valuable clues on the direction the field can take.