  Abelian gerbes and twisted bundles describe the topology of the NS-NS 3-form
gauge field strength H. We review how they have been usefully applied to study
and resolve global anomalies in open string theory. Abelian 2-gerbes and
twisted nonabelian gerbes describe the topology of the 4-form field strength G
of M-theory. We show that twisted nonabelian gerbes are relevant in the study
and resolution of global anomalies of multiple coinciding M5-branes. Global
anomalies for one M5-brane have been studied by Witten and by Diaconescu, Freed
and Moore. The structure and the differential geometry of twisted nonabelian
gerbes (i.e. modules for 2-gerbes) is defined and studied. The nonabelian
2-form gauge potential living on multiple coinciding M5-branes arises as
curving (curvature) of twisted nonabelian gerbes. The nonabelian group is in
general $\tilde\Omega E_8$, the central extension of the E_8 loop group. The
twist is in general necessary to cancel global anomalies due to the
nontriviality of the 11-dimensional 4-form G field strength and due to the
possible torsion present in the cycles the M5-branes wrap. Our description of
M5-branes global anomalies leads to the D4-branes one upon compactification of
M-theory to Type IIA theory.
