Flexible and multifunctional electronic devices and soft robots inspired by human organs, such as skin, have many applications. However, the emergence of electronic skins (e-skins) or textiles in biomedical engineering has made a great revolution in a myriad of people's lives who suffer from different types of diseases and problems in which their skin and muscles lose their appropriate functions. In this review, recent advances in the sensory function of the e-skins are described. Furthermore, we have categorized them from the sensory function perspective and highlighted their advantages and limitations. The categories are tactile sensors (including capacitive, piezoresistive, piezoelectric, triboelectric, and optical), temperature, and multi-sensors. In addition, we summarized the most recent advancements in sensors and their particular features. The role of material selection and structure in sensory function and other features of the e-skins are also discussed. Finally, current challenges and future prospects of these systems towards advanced biomedical applications are elaborated.