We recall and partially improve four versions of smooth, non-abelian gerbes:
Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We
prove that all these four versions are equivalent, and so establish new
relations between interesting recent developments. Prominent partial results we
prove are a bijection between continuous and smooth non-abelian cohomology, and
an explicit equivalence between bundle gerbes and principal 2-bundles as
2-stacks.Comment: 65 pages, v2: minor corrections, new Corollary 7.2 about equivariant
gerbes; v3: again minor corrections; v3 is the final and published versio
