Explore the theoretical foundations of Lagrangian mean curvature flow in this comprehensive lecture delivered at the International Centre for Theoretical Sciences. Delve into the mathematical framework that governs how Lagrangian submanifolds evolve under mean curvature flow, examining the geometric and analytical properties that make this area of differential geometry particularly rich and complex. Learn about the fundamental concepts, key theorems, and current research directions in this specialized field that sits at the intersection of symplectic geometry and geometric analysis. Discover how Lagrangian mean curvature flow relates to broader themes in minimal surface theory and understand its applications in various mathematical contexts. Gain insights into the technical challenges and open problems that researchers face when studying the long-time behavior and singularity formation in these geometric flows. This lecture forms part of a larger program on geometry and analysis of minimal surfaces, providing essential theoretical background for understanding recent developments in the field.