This paper reviews the state of the art of artificial tactile sensing, with a particular focus on bio-hybrid and fully-biological approaches. To this aim, the study of physiology of the human sense of touch and of the coding mechanisms of tactile information is a significant starting point, which is briefly explored in this review. Then, the progress towards the development of an artificial sense of touch are investigated. Artificial tactile sensing is analysed with respect to the possible approaches to fabricate the outer interface layer: synthetic skin versus bio-artificial skin. With particular respect to the synthetic skin approach, a brief overview is provided on various technologies and transduction principles that can be integrated beneath the skin layer. Then, the main focus moves to approaches characterized by the use of bio-artificial skin as an outer layer of the artificial sensory system. Within this design solution for the skin, bio-hybrid and fully-biological tactile sensing systems are thoroughly presented: while significant results have been reported for the development of tissue engineered skins, the development of mechanotransduction units and their integration is a recent trend that is still lagging behind, therefore requiring research efforts and investments. In the last part of the paper, application domains and perspectives of the reviewed tactile sensing technologies are discussed.