Explore the fundamental concepts of manifolds, coordinates, and tensors in this graduate-level lecture from the Warsaw4PhD and GeoPlanet PhD schools. Delve into the intricacies of manifolds, examining their properties and significance in theoretical physics. Investigate scalars, vectors, and tensors on manifolds, gaining a deeper understanding of their mathematical representations and applications. Learn about coordinate transformations and their crucial role in describing physical phenomena. Presented by Mikołaj Korzyński from the Center for Theoretical Physics, Polish Academy of Sciences, this comprehensive lecture provides a solid foundation for advanced studies in theoretical physics and mathematics.