This thesis is an introduction to the fractional Sobolev spaces and the fractional Laplace operator. We define the fractional Sobolev spaces and give their properties by comparing them with the classical version of Sobolev spaces. After giving the motivation that comes from the random walk theory, we define the fractional Laplacian. We focus on the mean-value property of s-harmonic functions and get into details of extension and maximum principle of the weak solution of the Dirichlet problem for the fractional Laplacian. Afterall, we explain the regularity of the weak solution of the Dirichlet problem for the fractional Laplacian inside a domain and up to the boundary, respectively.