The clique partitioning problem is the problem of finding a partition of a complete graph into disjoint cliques that maximizes the sum of the costs of the edges within the cliques. This np-hard problem can be solved by means of integer linear programming. We present a state-of-the-art cutting plane algorithm for this problem. In particular, we use various classes of valid inequalities of the much-studied clique partitioning polytope in order to tighten the linear programming relaxation. We evaluate our algorithm on a wide range of benchmark instances and show that we obtain better bounds than those reported in the literature. Moreover, we utilize the cutting plane algorithm in a branch and cut algorithm. This algorithm can compute optimal solutions and prove their optimality significantly faster than previous methods. The complete source code is publicly available at https://github.com/JannikIrmai/clique-partitioning .