  We formulate the theory of a 2-form gauge field on a Euclidean spacetime
lattice. In this approach, the fundamental degrees of freedom live on the faces
of the lattice, and the action can be constructed from the sum over Wilson
surfaces associated with each fundamental cube of the lattice. If we take the
gauge group to be $U(1)$, the theory reduces to the well-known abelian gerbe
theory in the continuum limit. We also propose a very simple and natural
non-abelian generalization with gauge group $U(N) \times U(N)$, which gives
rise to $U(N)$ Yang-Mills theory upon dimensional reduction. Formulating the
theory on a lattice has several other advantages. In particular, it is possible
to compute many observables, such as the expectation value of Wilson surfaces,
analytically at strong coupling and numerically for any value of the coupling.
