Skin is the largest organ of the human body, protecting it against the external environment. Despite high self-regeneration potential, severe skin defects will not heal spontaneously and need to be covered by skin substitutes. Tremendous progress has been made in the field of skin tissue engineering, in recent years, to develop new skin substitutes. Among them, hydrogels are one of the candidates with most potential to mimic the native skin microenvironment, due to their porous and hydrated molecular structure. They can be applied as a permanent or temporary dressing for different wounds to support the regeneration and healing of the injured epidermis, dermis, or both. Based on the material used for their fabrication, hydrogels can be subdivided into two main groups—natural and synthetic. Moreover, hydrogels can be reinforced by incorporating nanoparticles to obtain “in situ” hybrid hydrogels, showing superior properties and tailored functionality. In addition, different sensors can be embedded in hydrogel wound dressings to provide real-time information about the wound environment. This review focuses on the most recent developments in the field of hydrogel-based skin substitutes for skin replacement. In particular, we discuss the synthesis, fabrication, and biomedical application of novel “smart” hydrogels.