Topology morphing insulation enables the on-demand switching of thermal properties between insulative and conducting states through shape change. The adaptive nature of these systems allows them to regulate heat transfer by dynamically altering insulation materials or systems in response to changing conditions, including environmental factors, electrical grid dynamics, and occupant requirements. In this article, we highlight the potential of topology morphing insulation for advancing building envelope design, improving energy efficiency, and facilitating on-demand adjustments in effective thermal conductivity. We provide a comprehensive overview of topology morphing insulation, delving into its underlying principles, mechanisms, and potential applications. This review explores cutting-edge research and the potential application of insights from non-building concepts, such as nature, textiles, and origami. Additionally, it examines crucial aspects such as actuation mechanisms, effectiveness, lifecycle considerations, sustainability implications, and manufacturing feasibility. We discuss the potential benefits and challenges associated with implementing topology morphing insulation solutions. Thanks to its transformative capabilities, topology morphing insulation holds tremendous promise for advancing building envelope design, driving energy efficiency improvements, and facilitating responsive changes in effective thermal conductivity.