In the last chapter, it was discussed that the algebraic Riccati equation (ARE) need be solved in order to obtain the state-space solutions of the normalized coprime factorizations. In Chap. 2, the Lyapunov equation was employed to determine the controllability and observability gramians of a system. Both the algebraic Riccati and Lyapunov equations play prominent roles in the synthesis of robust and optimal control as well as in the stability analysis of control systems. In fact, the Lyapunov equation is a special case of the ARE. The ARE indeed has wide applications in control system analysis and synthesis. For example, the state-space formulation for particular coprime factorizations with a J-lossless (or dual J-lossless) numerator requires solving an ARE; in turn, the J-lossless and dual J-lossless systems are essential in the synthesis of robust controllers using the CSD approach. In this chapter, the ARE will be formally introduced. Solution procedures to AREs and their various properties will be discussed. Towards the end of this chapter, the coprime factorization approach to solve several spectral factorization problems is to be considered.