Polymeric materials have found increasing use in biomedical applications in the last decades. Among them, hydrogels represent the chosen class of materials to use in this field, in particular as wound dressings. They are generally non-toxic, biocompatible, and biodegradable, and they can absorb large amounts of exudates. Moreover, hydrogels actively contribute to skin repair promoting fibroblast proliferation and keratinocyte migration, allowing oxygen to permeate, and protecting wounds from microbial invasion. As wound dressing, stimuli-responsive systems are particularly advantageous since they can be active only in response to specific environmental stimuli (such as pH, light, ROS concentration, temperature, and glucose level). In this review, we briefly resume the human skin’s structure and functions, as well as the wound healing phases; then, we present recent advances in stimuli-responsive hydrogels-based wound dressings. Lastly, we provide a bibliometric analysis of knowledge produced in the field.