Recently, the fractional Schrödinger equation has attracted great attention from both physicists and mathematicians. It is a nonlocal equation with its nonlocality described by the fractional Laplacian -- an infinitesimal generator of a symmetric $\alpha$-stable Lévy process. The nonlocality of the fractional Laplacian introduces great challenges in both analysis and simulations of the fractional Schrödinger equation. In this talk, we will study the stationary states and dynamics of the fractional Schrödinger equation and compare them with those of the traditional Schrödinger equation. On the one hand, we will study the ground states of the fractional Schrödinger equation in various external potentials. In particular, numerical methods for discretizing the fractional Laplacian will be discussed. On the other hand, we will present the dynamics of plane wave and soliton solutions of the fractional Schrödinger equation. Stability analysis of the split-step method for simulating the fractional Schrödinger equation will be discussed.