Compressed sensing technology, especially the compressed speech sensing technology has gradually become the research hotspot in signal processing. The currently key issues of compressed speech sensing include the construction of the sparse decomposition matrix, the selection of the measurement matrix and the design of the reconstruction algorithm for speech signal. The important representatives of sparse decomposition matrix are the orthogonal basis, linear prediction matrix based on speech characteristics and overcomplete dictionary. For measurement matrix, the performance of reconstructed speech signals based on random measurement matrix is analyzed. For reconstruction algorithm, the robust reconstruction algorithms with noisy measurement or noisy speech signal are researched. In the paper, the above three kinds of compressed speech sensing technologies are introduced and compared, and the main applications of compressed speech sensing are also provided. Finally, the possible future research points of compressed speech sensing are discussed.