The Basis Pursuit (BP) problem consists in finding a least `1 norm solution of the underdetermined linear system Ax = b. It arises in many areas of electrical engineering and applied mathematics. Applications include signal compression and modeling, estimation, fitting, and compressed sensing. In this paper, we explore methods for solving the BP in a distributed environment, i.e., when the computational resources and the matrix A are distributed over several interconnected nodes. Special instances of this distributed framework include sensor networks and distributed memory and/or processor platforms. We consider two distribution paradigms: either the columns or the rows of A are distributed across the nodes. The several algorithms that we present impose distinct requirements on the degree of connectivity of the network and the per-node computational complexity.