This chapter investigates the distributed composite optimization problem over an undirected network with equality constraints. The optimization problem includes a smooth term and two possibly nonsmooth terms. Most existing results are devoted to developing distributed algorithms in a synchronous setting with a global clock, where the agents cannot move to the next iteration until the slowest agent completes its update. A new asynchronous distributed primal-dual forward-backward splitting algorithm (AD-PDFBS) is presented to solve this problem. Each agent can compute and communicate independently at different times, with the information it has even if the latest information from its neighbors is not yet available. The proposed algorithm is characterized by its ability to reduce synchronization wait and alleviate communication bottleneck. The convergence of AD-PDFBS is proved by transforming the asynchronous algorithm into a fixed-point problem utilizing the operator splitting scheme under bounded delay assumption. Experimental results confirm the effectiveness of AD-PDFBS.