Summary: Up to overall harmonic factors, the D8-brane solution of the massive type IIA supergravity theory is the product of nine-dimensional Minkowski space (the world-volume) with the real line (the transverse space). We show that the equations of motion allow for the world-volume metric to be generalised to an arbitrary Ricci-flat one. If this nine-dimensional Ricci-flat manifold admits Killing spinors, then the resulting solutions are supersymmetric and satisfy the usual Bogomol’nyi bound, although they preserve fewer than the usual one half of the supersymmetries. We describe the possible choices of such manifolds, elaborating on the connection between the existence of Killing spinors and the self-duality condition on the curvature two-form. Since the D8-brane is a domain wall in ten dimensions, we are led to consider the general case: domain walls in any supergravity theory. Similar considerations hold here also. Moreover, it is shown that the world-volume of any magnetic brane – of which the domain walls are a specific example – can be generalised in precisely the same way. The general class of supersymmetric solutions have gravitational instantons as their spatial sections. Some mention is made of the world-volume solitons of such branes.