Dr. Ganpathy Murthy University of Kentucky We thought we knew all there was to know about band insulators back in the 1930s. However, in the last 10 years we have learnt that there distinct types of band insulators in 2 and 3 dimensions. The distinction between these types is "topological", a term I will explain. I will introduce the idea of band topology in detail in 2D. I will then use the example of the integer quantum Hall effects to show that a topological insulator has edge states that are robust to disorder. Next I will introduce time-reversal invariance, which puts powerful constraints on band insulators. Once again, edge modes will prove to be extremely useful in characterizing the different types of band insulators. I will end up by talking about 3D topological insulators and some of the phenomenology associated with them.