Fractional diffusion equations are naturally derived on unbounded domains, and their solutions usually decay very slowly at infinity. A usual approach to dealing with unbounded domains is to use a domain truncation with exact or approximate transparent boundary conditions. But since accurate transparent boundary conditions at truncated boundaries are not easily available, we develop in this chapter efficient spectral methods for FPDEs on unbounded domains so as to avoid errors introduced by domain truncation. Formulation of Laplacians in bounded domains will be presented in Chapter 6.