This paper introduces several related distributed algorithms, generalised from the celebrated belief propagation algorithm for statistical learning. These algorithms are suitable for a class of computational problems in large-scale networked systems, ranging from average consensus, sensor fusion, distributed estimation, distributed optimisation, distributed control, and distributed learning. By expressing the underlying computational problem as a sparse linear system, each algorithm operates at each node of the network graph and computes iteratively the desired solution. The behaviours of these algorithms are discussed in terms of the network graph topology and parameters of the corresponding computational problem. A number of examples are presented to illustrate their applications. Also introduced is a message-passing algorithm for distributed convex optimisation.