This article presents novel results concerning the stability and robustness of recovery of signals from undersampled data by lq minimization where 0< q≤ 1. We show that the null space property (NSP)–a condition that is equivalent to exact recovery of any sparse signal from its noiseless observation–is sufficient to stably and robustly recover any approximately sparse signal via lq minimization. We further show that recovery of a signal via lq minimization is robust with respect to perturbation of the measurement matrix. Finally, we generalize lq minimization to F-minimization and introduce the corresponding NSP.