The application of multiple-input multiple-output (MIMO) techniques to nonorthogonal multiple access (NOMA) systems is important to enhance the performance gains of NOMA. In this paper, a novel MIMO-NOMA framework for downlink and uplink transmission is proposed by applying the concept of signal alignment. By using stochastic geometry, closed-form analytical results are developed to facilitate the performance evaluation of the proposed framework for randomly deployed users and interferers. The impact of different power allocation strategies, namely fixed power allocation and cognitive radio inspired power allocation, on the performance of MIMO-NOMA is also investigated. Computer simulation results are provided to demonstrate the performance of the proposed framework and the accuracy of the developed analytical results.