Abstract: In this talk I will review recent theoretical work on a new class of topological material systems - topological Kondo insulators, which appear as a result of interplay between strong correlations and spin-orbit interactions. I will start with introducing the by now standard theory of topological band insulators and explain the Fu-Kane method to calculate the Z2 topological index for time-reversal-invariant band structures in three dimensions. The method will be used to show that hybridization between the conduction electrons and localized f-electrons in certain heavy fermion compounds gives rise to interaction-induced topological insulating behavior. A mean field theory of these Kondo topological insulators will be derived. I will also discuss recent experimental results, which have conclusively confirmed our predictions in the Samarium hexaboride compound, where the long-standing puzzle of the residual low-temperature conductivity has been shown to originate from topological surface states. This material system represents the first true topological insulator observed experimentally with low-temperature transport dominated by the surface and essentially no conduction in the bulk. In conclusion, I will mention our ongoing theory work, which focuses on very unusual non-linear transport properties of Samarium hexaboride devices, which mimic neuron-like behavior in biological systems.