Physics has benefited greatly from ideas that were independently developed in mathematics. Topology is a branch of mathematics that studies shapes. Properties that do not change under continuous deformation including stretching and bending are topological in nature. Now some of these topological ideas have become important in several areas of physics. In this talk, I will begin with an introduction of topological phases of matter in free Fermionic systems. I will then focus on my work on topological phase transitions in a new and important two-dimensional Dirac material silicene. A combination of intrinsic spin-orbit interaction and an external electric field leads to quantum phase transitions at the charge neutrality point in silicene. This phase transition from a topological insulator to a trivial insulator phase in silicene is accompanied by a quenching of quantum spin Hall effect and onset of valley Hall effect. Possible experimental tunability of topological state of silicene will also be discussed.