This paper introduces Regularized Orthogonal Matching Pursuit (ROMP), a greedy algorithm for recovering sparse signals from incomplete and inaccurate measurements. ROMP combines the speed and simplicity of greedy methods with the strong theoretical guarantees of convex programming. It is shown that ROMP can reliably recover an n-sparse signal v from measurements x = Φv + e, where Φ is a measurement matrix and e is an error vector. Under the Restricted Isometry Condition, ROMP recovers the signal in at most n iterations, each involving a least squares problem. The reconstruction error is proportional to √(log n)‖e‖₂. If e is zero, the reconstruction is exact. ROMP is also stable under measurement perturbations and can recover approximately sparse signals. The algorithm is shown to be as stable as convex programming methods, with error bounds that depend on the error vector and the sparsity of the signal. The paper proves that ROMP can recover a signal v with high accuracy even when the measurements are noisy or the signal is not exactly sparse. It also shows that ROMP can be used to approximate the best n-term approximation of a signal, with error bounds that depend on the error vector and the sparsity of the signal. Numerical experiments demonstrate that ROMP performs well in practice, with recovery errors that are often better than the theoretical bounds.This paper introduces Regularized Orthogonal Matching Pursuit (ROMP), a greedy algorithm for recovering sparse signals from incomplete and inaccurate measurements. ROMP combines the speed and simplicity of greedy methods with the strong theoretical guarantees of convex programming. It is shown that ROMP can reliably recover an n-sparse signal v from measurements x = Φv + e, where Φ is a measurement matrix and e is an error vector. Under the Restricted Isometry Condition, ROMP recovers the signal in at most n iterations, each involving a least squares problem. The reconstruction error is proportional to √(log n)‖e‖₂. If e is zero, the reconstruction is exact. ROMP is also stable under measurement perturbations and can recover approximately sparse signals. The algorithm is shown to be as stable as convex programming methods, with error bounds that depend on the error vector and the sparsity of the signal. The paper proves that ROMP can recover a signal v with high accuracy even when the measurements are noisy or the signal is not exactly sparse. It also shows that ROMP can be used to approximate the best n-term approximation of a signal, with error bounds that depend on the error vector and the sparsity of the signal. Numerical experiments demonstrate that ROMP performs well in practice, with recovery errors that are often better than the theoretical bounds.