Abstarct: In this thesis, we study fractional optimal control problems and fractional infinite horizon optimal control problems. To solve these problems, we employ spectral and pseudo-spectral methods. For fractional optimal control problems, we use fractional Lagrange polynomials and approximate the solutions of the problems by fractional interpolation functions at shifted Gauss–Legendre points. For problems with an infinite horizon, we use a domain transformation technique to map the infinite domain to a finite one and then propose a shifted Legendre pseudo-spectral method to solve the resulting problem. Finally, we introduce a new fractional derivative operator called the adaptive derivative, which, unlike other common fractional derivatives (such as Caputo and Riemann–Liouville), possesses desirable properties such as the product rule and the chain rule, and has a direct relationship with the classical derivative for integer orders. The convergence of the proposed methods is proved under mild conditions and their high efficiency and accuracy are clearly demonstrated by presenting numerical examples and comparing them with existing methods.