Compressed sensing (CS) is widely used in different areas. The key technologies of compressed sensing include the selection of sparse matrix, the construction of the measurement matrix, and the design of the reconstruction algorithm. Speech signal usually has special structural characteristics in the measurement matrix and reconstruction algorithm. In actual applications, noises may inevitably exist. In compressed sensing theory, the reconstruction system is nonlinear and sensitive to noise. Therefore, we need to study the robust compressed sensing technology. This technique would have utilizable perspective, if the robustness problem gets solved. The paper begins with the concept of compressed sensing, then analyses the effects brought by various noises. When it comes to the solutions to the noises in the speech signal, this paper focuses on the introduction of robust projection operator and robust recovery algorithms. Finally, the possible future research directions are prospected.