The paper introduces NESTA, a fast and accurate algorithm for solving sparse recovery problems in signal processing, inspired by Nesterov's smoothing technique. NESTA is designed to efficiently handle large-scale compressed sensing reconstruction problems, offering computational efficiency, high accuracy, flexibility, and robustness. The algorithm's key idea is a subtle averaging of iterates, which improves convergence properties. It is particularly effective for problems with a wide dynamic range and can solve $\ell_1$ minimization and total-variation minimization problems. Numerical experiments demonstrate that NESTA outperforms state-of-the-art methods in terms of speed and accuracy, making it a valuable tool for signal recovery from indirect and undersampled data. The paper also discusses the use of continuation techniques to accelerate convergence and provides theoretical insights into the algorithm's performance.The paper introduces NESTA, a fast and accurate algorithm for solving sparse recovery problems in signal processing, inspired by Nesterov's smoothing technique. NESTA is designed to efficiently handle large-scale compressed sensing reconstruction problems, offering computational efficiency, high accuracy, flexibility, and robustness. The algorithm's key idea is a subtle averaging of iterates, which improves convergence properties. It is particularly effective for problems with a wide dynamic range and can solve $\ell_1$ minimization and total-variation minimization problems. Numerical experiments demonstrate that NESTA outperforms state-of-the-art methods in terms of speed and accuracy, making it a valuable tool for signal recovery from indirect and undersampled data. The paper also discusses the use of continuation techniques to accelerate convergence and provides theoretical insights into the algorithm's performance.