This book presents the state-of-the-art in theory and algorithms for signal recovery under the sparsity assumption. The unique conditions for the sparsest solution of underdetermined linear systems are described, and the results for sparse signal recovery under the range space property (RSP) are introduced. This framework is generalized to 1-bit compressed sensing, leading to a novel sign recovery theory in this area. Two efficient sparsity-seeking algorithms are presented, and theoretical efficiency of these algorithms are rigorously analysed. Under the RSP assumption, the author also provides a unified stability analysis for several popular optimization methods for sparse signal recovery.