Inspired by the advantages of the distributed network structure in large-scale data processing and system robustness, this chapter considers a fully distributed computation strategy for a nonsmooth composite optimization problem. The goal is to solve the problem only by resorting to local computation and local communications. Here, the local objective function is composed of two possibly nonsmooth functions where one is strongly convex and the other is proper closed convex (not necessarily strongly convex). Through associating variables with the agents and edges of the network, a distributed synchronous algorithm is developed based on the dual operator splitting method. The asynchronous version of the proposed algorithm is further developed based on the randomized block coordinate descent technique, where each agent performs the algorithm only under its independent activation. Then the convergence analysis is provided depending on the fixed point iterations and the property of strong duality. Simulations on regularized problems demonstrate the effectiveness of the proposed distributed algorithm.