This paper surveys major practical algorithms for sparse approximation, focusing on computational issues, performance in different scenarios, and theoretical guarantees. Sparse approximation aims to approximate a target signal using a linear combination of a few elementary signals from a fixed collection. This problem is relevant in many areas, including signal processing, statistics, and applied mathematics. The paper discusses five major algorithmic approaches: greedy pursuit, convex relaxation, Bayesian framework, nonconvex optimization, and brute force. Greedy pursuit methods, such as orthogonal matching pursuit (OMP) and compressive sampling matching pursuit (CoSaMP), are computationally practical and yield provably correct solutions under certain conditions. Convex relaxation methods, such as those based on $ \ell_1 $-norm optimization, are also discussed, along with their theoretical guarantees. The paper also covers iterative thresholding methods and their performance in sparse approximation. It highlights the importance of the restricted isometry property (RIP) in ensuring the success of these algorithms. The paper concludes with a discussion of the advantages and limitations of different approaches, emphasizing the role of greedy algorithms and convex optimization in sparse approximation.This paper surveys major practical algorithms for sparse approximation, focusing on computational issues, performance in different scenarios, and theoretical guarantees. Sparse approximation aims to approximate a target signal using a linear combination of a few elementary signals from a fixed collection. This problem is relevant in many areas, including signal processing, statistics, and applied mathematics. The paper discusses five major algorithmic approaches: greedy pursuit, convex relaxation, Bayesian framework, nonconvex optimization, and brute force. Greedy pursuit methods, such as orthogonal matching pursuit (OMP) and compressive sampling matching pursuit (CoSaMP), are computationally practical and yield provably correct solutions under certain conditions. Convex relaxation methods, such as those based on $ \ell_1 $-norm optimization, are also discussed, along with their theoretical guarantees. The paper also covers iterative thresholding methods and their performance in sparse approximation. It highlights the importance of the restricted isometry property (RIP) in ensuring the success of these algorithms. The paper concludes with a discussion of the advantages and limitations of different approaches, emphasizing the role of greedy algorithms and convex optimization in sparse approximation.