  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.
