Fan-planar graphs seem to provide a suitable graph-theoretical foundation for edge bundling which is heavily being used for visualization purposes. We apply the fan-planarity concept to edge bundles and introduce the model of fan-bundle-planarity. For the restricted one-sided variant where each edge is crossed by at most one bundle and which is a special case of fan-planarity, we give a broad range of results, from recognition to edge density, from outer-fan-bundle-planarity to the 2-layer variant. For the more natural and general two-sided variant where each edge might be part of bundles with both its end segments, i.e. two bundles, we present preliminary results, observations and conjectures.