Non-abelian gerbes are a generalization of principal G-bundles, involving the replacement of the Lie group G by a Lie 2-group, or crossed module of groups, not necessarily Abelian. Apart from providing a nice example of categorification in geometry, they have found a number of applications in physics, e.g. in higher gauge theory and topological states of matter.

I will describe the geometry of non-abelian gerbes, both in terms of local connection 1- and 2-forms and curvature 3-forms, and in terms of transports along paths and surfaces (2-paths). There is a notion of gauge equivalence in both cases. This part is based on work with João Faria Martins.

As for principal G-bundles, we would like to understand the moduli space of flat gerbe connections modulo gauge transformations. I will present work in progress with Jeffrey Morton, where we categorify the action of a group on a set to obtain the action of a 2-group on a category, and apply this to the action of the 2-group of gauge transformations on the category of connections.

Finally I will describe some results obtained with Diogo Bragança, for the case when the 2-group is finite. Here we can define a counting invariant for a class of surfaces with boundary, which can be interpreted as the volume of the corresponding moduli space, in a suitable sense.