Graham P. Smith, Tessa Baker, Simon Birrer, Christine E. Collins, Jose María Ezquiaga, Srashti Goyal, Otto A. Hannuksela, Phurailatpam Hemantakumar, Martin A. Hendry, Justin Janquart, David Keitel, Andrew J. Levan, Rico K. L. Lo, Anupreeta More, Matt Nicholl, Inés Pastor-Marazuela, Andrés I. Ponte Pérez, Helena Ubach, Laura E. Uronen, Mick Wright, Miguel Zumalacarregui, Federica Bianco, Mesut Çalışkan, Juno C. L. Chan, Elena Colangeli, Benjamin P. Gompertz, Christopher P. Haines, Erin E. Hayes, Bin Hu, Gavin P. Lamb, Anna Liu, Soheb Mandhai, Harsh Narola, Quynh Lan Nguyen, Jason S. C. Poon, Dan Ryczanowski, Eungwang Seo, Anowar J. Shajib, Xikai Shan, Nial Tanvir, Luka Vujeva

We introduce the rapidly emerging field of multi-messenger gravitational lensing - the discovery and science of gravitationally lensed phenomena in the distant universe through the combination of multiple messengers. This is framed by gravitational lensing phenomenology that has grown since the first discoveries in the 20th century, messengers that span 30 orders of magnitude in energy from high energy neutrinos to gravitational waves, and powerful "survey facilities" that are capable of continually scanning the sky for transient and variable sources. Within this context, the main focus is on discoveries and science that are feasible in the next 5-10 years with current and imminent technology including the LIGO-Virgo-KAGRA network of gravitational wave detectors, the Vera C. Rubin Observatory, and contemporaneous gamma/X-ray satellites and radio surveys. The scientific impact of even one multi-messenger gravitational lensing discovery will be transformational and reach across fundamental physics, cosmology and astrophysics. We describe these scientific opportunities and the key challenges along the path to achieving them. This article is the introduction to the Theme Issue of the Philosophical Transactions of The Royal Society A on the topic of Multi-messenger Gravitational Lensing, and describes the consensus that emerged at the associated Theo Murphy Discussion Meeting in March 2024.