On-shell and conformal N=4 supergravity in superspace

We discuss the superspace geometries which are necessary to describe on-shell O(4) and SU(4) supergravity. The relation of central charge field strengths to physical spin-zero fields is exhibited and a 'new' O(4) theory is shown to exist. The version of SU(4) supergravity which uses and antisymmetric tensor gauge field is found to require modifications of ordinary superspace. Finally the Poincare supergeometry which admits the conformal N=4 supermultiplet is constructed. It is shown that consistency of the Bianchi identities implies the existence of dimension zero auxiliary fields which are components of a non-linear multiplet. (orig.)