The deepening integration of social-technical systems creates immensely complex environments, creating increasingly uncertain and unpredictable circumstances. Given this context, policymakers have been encouraged to draw on complexity science-informed approaches in policymaking to help grapple with and manage the mounting complexity of the world. For nearly eighty years, complexity-informed approaches have been promising to change how our complex systems are understood and managed, ultimately assisting in better policymaking. Despite the potential of complexity science, in practice, its use often remains limited to a few specialised domains and has not become part and parcel of the mainstream policy debate. To understand why this might be the case, we question why complexity science remains nascent and not integrated into the core of policymaking. Specifically, we ask what the non-technical challenges and barriers are preventing the adoption of complexity science into policymaking. To address this question, we conducted an extensive literature review. We collected the scattered fragments of text that discussed the non-technical challenges related to the use of complexity science in policymaking and stitched these fragments into a structured framework by synthesising our findings. Our framework consists of three thematic groupings of the non-technical challenges: (a) management, cost, and adoption challenges; (b) limited trust, communication, and acceptance; and (c) ethical barriers. For each broad challenge identified, we propose a mitigation strategy to facilitate the adoption of complexity science into policymaking. We conclude with a call for action to integrate complexity science into policymaking further.