The idea behind topological systems is simple: if there exists a quantity, which cannot change in an insulating system where all the particles are localized, then the system must become conducting and obtain propagating particles, when this quantity (called "topological invariant") finally changes. Frequently, the edges of such topological materials have properties that are impossible to achieve otherwise owing to the so-called "bulk-edge correspondence". It guarantees the existence of protected states at the edge and their robustness against anything that happens at the boundary. The practical applications of this principle are quite profound, and already within the last eight years they have lead to prediction and discovery of a vast range of new materials with exotic properties that were considered to be impossible before. Our central focus will be these very exciting developments with special attention to the most active research topics in topological condensed matter: namely the theory of topological insulators and superconductors following from the 'grand ten symmetry classes' as well as topological quantum computation and Majoranas. We will complete this general picture with a discussion of some of the other ramifications of topology in various areas of condensed matter such as photonic and mechanical systems, topological quantum walks, topology in fractionalized systems, driven or dissipative systems. We aim to allow the people taking the course to achieve three objectives: Learn about the variety of subtopics in topological materials, their relation to each other and to the general principles. Learn to follow active research on topology, and critically understand it on your own. Acquire skills required to engage in research on your own, and to minimize confusion that often arises even among experienced researchers. The basic tool that we are going to use in the course is going to be simple thought experiments that rely on considerations of symmetry or continuity under adiabatic deformations. We are then going to look at the most relevant research papers and teach how to simply understand the idea even in the rather involved ones. Finally, in order to give a more detailed and visual understanding of the involved concepts we are also going to use computer simulations similar to those used in actual research. This course is a joint effort of Delft University of Technology, QuTech, and University of Maryland. LICENSEThe course materials of this course are Copyright Delft University of Technology and are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike (CC-BY-NC-SA) 4.0 International License.