  We obtain singularity resolutions for various overlapping brane
configurations, including those of two heterotic 5-branes, type II 5-branes or
D4-branes. In these solutions, the ``harmonic'' function H for each brane
component depends only on the associated four-dimensional relative transverse
space. The resolution is achieved by replacing these transverse spaces with
Eguchi-Hanson or Taub-NUT spaces, both of which admit a normalisable self-dual
(or anti-self-dual) harmonic 2-form. Due to the manner in which the interaction
terms for the form fields modify their Bianchi identities or equations of
motion, these normalisable harmonic 2-forms provide regular sources for the
branes. We also obtain resolved 5-branes and D4-branes wrapped on S^1, which is
fibred over the transverse Eguchi-Hanson or Taub-NUT spaces. The T-duality
invariance of the NS-NS 5-brane is retained after the resolution. The resolved
5-branes and D4-branes provide regular supergravity duals of certain
supersymmetric Yang-Mills theories in five and four dimensions.
