This paper focuses on studying the fundamental performance limits and linear dispersion code design for the MIMO-ARQ slow fading channel. The optimal average rate of well-known HARQ protocols is analyzed. The optimal design of space-time coding for the MIMO-ARQ channel is discussed. Information-theoretic measures are used to optimize the rate assignment and derive the optimum design criterion, which is then used to evaluate the optimality of existing space-time codes. A different design criterion, which is obtained from the error probability analysis of space-time coded MIMO-HARQ, is presented. Examples are studied to reveal the gain of ARQ feedback in space-time coded MIMO systems.