The maximum clique problem is a fundamental challenge in combinatorial optimization with significant applications across various fields. This paper provides a comprehensive survey of algorithms, computational complexity, and applications related to the maximum clique problem. It covers both exact and heuristic methods, including integer programming formulations, continuous formulations, enumerative algorithms, and various heuristics such as sequential greedy, local search, simulated annealing, neural networks, genetic algorithms, tabu search, and continuous-based heuristics. The paper also discusses specific applications in coding theory, tiling geometry, fault diagnosis, computer vision, and pattern recognition. The authors build upon existing literature and provide an updated bibliography to offer a thorough overview of the current state of research in this area.The maximum clique problem is a fundamental challenge in combinatorial optimization with significant applications across various fields. This paper provides a comprehensive survey of algorithms, computational complexity, and applications related to the maximum clique problem. It covers both exact and heuristic methods, including integer programming formulations, continuous formulations, enumerative algorithms, and various heuristics such as sequential greedy, local search, simulated annealing, neural networks, genetic algorithms, tabu search, and continuous-based heuristics. The paper also discusses specific applications in coding theory, tiling geometry, fault diagnosis, computer vision, and pattern recognition. The authors build upon existing literature and provide an updated bibliography to offer a thorough overview of the current state of research in this area.