Multiple input multiple output (MIMO) radar exhibits several advantages with respect to traditional radar array systems in terms of flexibility and performance. However, MIMO radar poses new challenges for both hardware design and digital processing. In particular, achieving high azimuth resolution requires a large number of transmit and receive antennas. In addition, the digital processing is performed on samples of the received signal, from each transmitter to each receiver, at its Nyquist rate, which can be prohibitively large when high resolution is needed. Overcoming the rate bottleneck, sub-Nyquist sampling methods have been proposed that break the link between radar signal bandwidth and sampling rate. In this work, we extend these methods to MIMO configurations and propose a sub-Nyquist MIMO radar (SUMMeR) system that performs both time and spatial compression. We present a range-azimuth-Doppler recovery algorithm from sub-Nyquist samples obtained from a reduced number of transmitters and receivers, that exploits the sparsity of the recovered targets' parameters. This allows us to achieve reduction in the number of deployed antennas and the number of samples per receiver, without degrading the time and spatial resolutions. Simulations illustrate the detection performance of SUMMeR for different compression levels and shows that both time and spatial resolution are preserved, with respect to classic Nyquist MIMO configurations. We also examine the impact of design parameters, such as antennas' locations and carrier frequencies, on the detection performance, and provide guidelines for their choice.