Millimeter wave automotive radar is one of the most important sensors in the advanced driver assistance system(ADAS) and autonomous driving system，which detects the target vehicles around the ego vehicle via processing transmitted and echo signals. However, the classical radar signal processing methods’ sampling frequency is very high, because the Shannon Nyquist sampling theorem which requires the sampling frequency of IF signal must be greater than the double of IF signal frequency is used to determine the sampling frequency of the IF signal. Besides, for the classical radar signal processing methods, the resolution of range, velocity and azimuth can’t meet the requirement of highly automated driving, especially azimuth. In order to solve these problems, a new novel MIMO(multiple-input multiple-output) radar target detection algorithm based on compressive sensing in the spatial domain is proposed in this paper. Because the radar target vehicles are highly sparse in the whole space background, therefore in the 3D space of range-doppler-azimuth, radar target vehicles are also highly sparse. So the problem of target detection of range, velocity and azimuth can be turned into a sparse vector-reconstruction problem of compressive sensing. In this paper, firstly target vehicles echo signal model of MIMO radar is established. And then the radar RF front end and echo signal are modelled in SystemVue. Finally, radar signal processing using compressive sensing is taken place in MATLAB. This signal processing algorithm includes three parts: sparse representation and linear measurement of echo signal and sparse signal reconstruction by using Orthogonal Matching Pursuit algorithm (OMP). The simulation results show that the proposed signal processing algorithm in this paper using compressive sensing technique in automotive MIMO radar can effectively reduce the sampling frequency and the resolution of range, velocity and azimuth is improved compared with the classical radar signal processing algorithm.