Explore the fascinating world of integrable vertex models in this comprehensive lecture presented by Leonid Petrov from the University of Virginia. Delve into classical and recent developments, focusing on applications to random growth and particle systems. Discover the connections between integrable vertex models and random tilings, symmetric functions, and other related structures. Gain insights into the intersection of geometry, statistical mechanics, and integrability as part of IPAM's Geometry, Statistical Mechanics, and Integrability Tutorials series. Recorded on March 15, 2024, at the Institute for Pure & Applied Mathematics (IPAM) at UCLA, this 68-minute talk offers a deep dive into advanced mathematical concepts and their applications.