This paper is devoted to the derivation and analysis of accurate and efficient Perfectly Matched Layers (PML) or efficient absorbing layers for solving fractional Laplacian equations within Initial Boundary Value Problems (IBVP). Two main approaches are derived: we first propose a Fourier-based pseu-dospectral method, and then present a real space method based on an efficient computation of the fractional Laplacian with PML. Some numerical experiments and analytical results are proposed along the paper to illustrate the presented methods.