This paper considers the application of multiple-input multiple-output (MIMO) techniques to nonorthogonal multiple access (NOMA) systems. A new design of precoding and detection matrices for MIMO-NOMA is proposed and its performance is analyzed for the case with a fixed set of power allocation coefficients. To further improve the performance gap between MIMO-NOMA and conventional orthogonal multiple access schemes, user pairing is applied to NOMA and its impact on the system performance is characterized. More sophisticated choices of power allocation coefficients are also proposed to meet various quality-of-service requirements. Finally, computer simulation results are provided to facilitate the performance evaluation of MIMO-NOMA and also demonstrate the accuracy of the developed analytical results.