In this talk I will review theoretical and experimental 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. Next, I will discuss recent experimental results, which confirm 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. Samarium hexaboride is perhaps the only material currently available, where low-temperature transport is truly surface-dominated and has essentially no conduction in the bulk. I will also discuss recent non-perturbative results on topological Kondo insulators in low-dimensional systems. In conclusion, I will mention our ongoing theory-experimental collaborative work, which has resulted in creation of new, practical devices based on the topological surface states in Samarium hexaboride.