Prof Shivaji Sondhi explains how topology is applied to understanding properties of condensed matter systems, providing an introduction to topics including defects & solitons, the quantum Hall effect, and topological insulators. The mathematics of topology has been applied with increasing success to understanding the properties of condensed matter systems. Indeed, it is fair to say that over the past couple of decades, it has revolutionized our understanding of what forms of order are possible in quantum matter. In this talk, I will provide a brief, pedagogical introduction to some of the most well-known applications of topological ideas. These will include topics such as defects & solitons, the quantum Hall effect, and topological insulators.