What should we think to be a non-abelian 2-form?

I will report on my research regarding generalization of non-abelian integration to the dimension two (i.e. over surfaces). The non-abelian analogs of 2-forms (objects admitting invariant integration) and their gauge symmetries will be described. I will try to explain the essence of integration procedure in this framework and illustrate it with examples "occuring in nature", including the abelian limit and Lie algebra -> Lie group integration. If time permits, we can also discuss which difficulties are to overcome to write down the explicit formula for 2d integrals, and, independently, the reasons I consider this construction to be more general than the existing ones (based e.g. on crossed modules or some of their friends).