In this paper we develop an approach to designing reduced-order multirate controllers. A discrete-time model that accounts for the multirate timing sequence of measurements is presented and is shown to have periodically time-varying dynamics. Using discrete-time stability theory, the optimal projection approach to fixed-order (i.e., full- and reduced-order) dynamic compensation is generalized to obtain reduced-order periodic controllers that account for the multirate architecture. It is shown that the optimal reduced-order controller is characterized by means of a periodically time-varying system of equations consisting of coupled Riccati and Lyapunov equations. In addition, the multirate static output-feedback control problem is considered. For both problems, the design equations are presented in a concise, unified manner to facilitate their accessibility for developing numerical algorithms for practical applications. Finally, a novel homotopy algorithm, based on a prediction and a Newton correction scheme, is also presented which allows solutions to periodic difference Riccati equations.