The statistical dependent source separation problem is a basic and important research topic in the field of blind source separation(BSS), because it not only has abundant potential applications, but also can gain further insights into the structure of the data. Firstly, the unified mathematical model of dependent source separation is constructed and the separability issues are discussed. Then, state-of-art algorithms to implement separation for dependent sources are surveyed from two aspects: statistical and deterministic, where the statistical method mainly includes sparsity based method, statistical measure based method, independent subspace analysis(ISA) based method and temporal structure based method. Meanwhile, the deterministic approach contains the bounded source signals based method and nonnegative source signals based method. The applications of the statistical dependent source separation problem are demonstrated by the hyperspectral unmixing problem and the BSS in the additive noise problem. Finally, some of the existing problems are listed, and the future research work is also presented.