Bioprinting technologies, which have the ability to combine various human cell phenotypes, signaling proteins, extracellular matrix components, and other scaffold-like biomaterials, are currently being exploited for the fabrication of human skin in regenerative medicine. We performed a systematic review to appraise the latest advances in 3D bioprinting for skin applications, describing the main cell phenotypes, signaling proteins, and bioinks used in extrusion platforms. To understand the current limitations of this technology for skin bioprinting, we briefly address the relevant aspects of skin biology. This field is in the early stage of development, and reported research on extrusion bioprinting for skin applications has shown moderate progress. We have identified two major trends. First, the biomimetic approach uses cell-laden natural polymers, including fibrinogen, decellularized extracellular matrix, and collagen. Second, the material engineering line of research, which is focused on the optimization of printable biomaterials that expedite the manufacturing process, mainly involves chemically functionalized polymers and reinforcement strategies through molecular blending and postprinting interventions, i.e., ionic, covalent, or light entanglement, to enhance the mechanical properties of the construct and facilitate layer-by-layer deposition. Skin constructs manufactured using the biomimetic approach have reached a higher level of complexity in biological terms, including up to five different cell phenotypes and mirroring the epidermis, dermis and hypodermis. The confluence of the two perspectives, representing interdisciplinary inputs, is required for further advancement toward the future translation of extrusion bioprinting and to meet the urgent clinical demand for skin equivalents.