  Bundle gerbes are simple examples of higher geometric structures that show
their utility in dealing with topological subtleties of physical theories. I
review a recent construction of torsion topological invariants for condensed
matter systems via equivariant bundle gerbes. The construction covers static
and periodically driven systems with time reversal invariance in 2 and 3 space
dimensions. It involves refinements of geometry of gerbes that are discussed in
the first lecture, the second one being devoted to the applications to
topological insulators.
