The maximum clique problem is a well-studied challenge in graph theory with applications in network analysis, bioinformatics, and social network modeling. In this senior integrative exercise for the Carleton College Computer Science Department, we implemented four algorithms for solving the problem, comparing their efficiency, accuracy, and scalability. We examine both exact and approximate approaches, discussing their computational complexity and practical applications. Experimental results are presented to evaluate performance across different graph structures. Through this analysis, we highlight the trade-offs between optimality and computational feasibility, providing insight into the strengths and limitations of these algorithms.