We propose a class of field theories featuring solitonic solutions in which topological defects can end when they intersect other defects of equal or higher dimensionality. Such configurations may be termed ``Dirichlet topological defects'', in analogy with the D-branes of string theory. Our discussion focuses on defects in scalar field theories with either gauge or global symmetries, in (3+1) dimensions; the types of defects considered include walls ending on walls, strings on walls, and strings on strings.

13 pages, 3 figures, RevTeX. (References improved.)