Explore the fascinating world of topological phases in quantum matter through this comprehensive lecture by Sumathi Rao at the International Centre for Theoretical Sciences. Delve into the fundamental concepts of topology and its applications in condensed matter physics, starting with the groundbreaking discovery of the integer quantum Hall effect. Examine the bulk-boundary correspondence and edge states, and learn about the recent developments in topological materials, including topological insulators and superconductors. Investigate key models like the Su-Schrieffer-Heeger and Kitaev models, and gain insights into current research frontiers such as Floquet topological phases and the quest for Majorana modes. This in-depth presentation offers a thorough overview of the field, from its historical foundations to cutting-edge advancements, providing valuable knowledge for students, researchers, and professionals interested in quantum physics and materials science.