Born-Infeld theory admits finite energy point particle solutions with δ-function sources, Blons. I discuss their role in the theory of Dirichlet p-branes as the ends of strings intersecting the brane when the effects of gravity are ignored. There are also topologically non-trivial electrically neutral catenoidal solutions looking like two p-branes joined by a throat. The general solution is a non-singular deformation of the catenoid if the charge is not too large and a singular deformation of the BIon solution for charges above that limit. The intermediate solution of BPS and Coulomb-like. Performing a duality rotation we obtain monopole solutions, the BPS limit being a solution of the abelian Bogolmol'nyi equations. The situation closely resembles that of sub and super extreme black-brane solutions of the supergravity theories. I also show that certain special Lagrangian submanifolds of Cp, p = 3, 4, 5, may be regarded as supersymmetric configurations consisting of p-branes at angles joined by throats which are the sources of global monopoles. Vortex solutions are also exhibited.