Abstract: In the basic Compressed sensing (CS), the unknown sparse signal is recovered from a single measurement vector, this is referred to as a single measurement vector (SMV) model. But in many applications, we should recover the joint sparse source signals from a set of measurement vectors. This is called the multiple measurement vectors(MMV) problem of CS, which addresses the recovery of a set of sparse signal vectors that share common non-zero support. This paper begins with the basic mathematic model of SMV and MMV in detail, followed by the existences and uniqueness conditions of the solution to the SMV and MMV. Then, the algorithms treating MMV model are overviewed and analyzed in detail, which are divided into three classes: convex method, greedy method and Bayesian method. These algorithms mathematics frameworks and performances are especially analyzed. At last, the existing problems that need further research are pointed out and some current challenges and future trends are summed up and predicted.