This paper presents several L1-minimization algorithms for sparse signal reconstruction based on a continuous-time projection neural network (PNN). First, a one-layer projection neural network is designed based on a projection operator and a projection matrix. The stability and global convergence of the proposed neural network are proved. Then, based on a discrete-time version of the PNN, several L1-minimization algorithms for sparse signal reconstruction are developed and analyzed. Experimental results based on random Gaussian sparse signals show the effectiveness and performance of the proposed algorithms. Moreover, experimental results based on two face image databases are presented that reveal the influence of sparsity to the recognition rate. The algorithms are shown to be robust to the amplitude and sparsity level of signals as well as efficient with high convergence rate compared with several existing L1-minimization algorithms.