Explore the fundamental limitations and breakdown conditions of the Migdal-Eliashberg theory in this 55-minute conference talk from the 2025 Simons Collaboration on New Frontiers in Superconductivity Annual Meeting. Delve into the theoretical framework that has been central to understanding conventional superconductivity and examine the specific circumstances under which this well-established theory fails to accurately describe superconducting phenomena. Analyze the mathematical foundations, physical assumptions, and parameter regimes where the Migdal-Eliashberg approach becomes inadequate, gaining insights into the ongoing challenges in superconductivity research and the need for alternative theoretical approaches in certain material systems and conditions.