Allen-Cahn equation arises in the mathematical study of phase transition. Despite its seemingly simple appearance, the equation has connections to minimal surfaces theory and displays a very rich structure involving deep mathematics. In this talk, I  will give a review of the history of the equation, and discuss the existence, symmetry and classification of finite Morse index solutions of the Allen-Cahn equation. In particular, I will report some recent results on the classification of Morse index solutions in low dimensions.