  p-Gerbes are a generalization of bundles that have (p+2)-form field
strengths. We develop their properties and use them to show that every theory
of p-gerbes can be reinterpreted as a gauge theory containing p-dimensional
extended objects. In particular, we show that every closed (p+2)-form with
integer cohomology is the field strength for a gerbe, and that every p-gerbe is
equivalent to a bundle with connection on the space of p-dimensional
submanifolds of the original space. We also show that p-gerbes are equivalent
to sheaves of (p-1)-gerbes, and use this to define a K-theory of gerbes. This
K-theory classifies the charges of (p+1)-form connections in the same way that
bundle K-theory classifies 1-form connections.
