Chapter 3 delves into the application of spectral methods for solving optimal control problems constrained by partial differential equations (PDEs). The chapter investigates two specific optimal control scenarios: optimal control of a fourth-order PDE with an \(H^1\) -norm state constraint and optimal flow control with an \(L^2\) -norm constraint on the control variable. The chapter begins by introducing the spectral approximation technique and proceeds to derive optimality conditions for each problem. Subsequently, rigorous a priori and a posteriori error estimates are established to assess the accuracy of the spectral approximations. The inclusion of a fourth-order parabolic optimal control problem further demonstrates the versatility and power of spectral methods in tackling a broad range of optimal control challenges.