  We study the different phases of field theories of compact antisymmetric
tensors of rank $h-1$ in arbitrary space-time dimensions $D=d+1$. Starting in a
`Coulomb' phase, topological defects of dimension $d-h-1$ ($(d-h-1)$-branes)
may condense leading to a generalized `confinement' phase. If the dual theory
is also compact the model may also have a third, generalized `Higgs' phase,
driven by the condensation of the dual $(h-2)$-branes. Developing on the work
of Julia and Toulouse for ordered solid-state media, we obtain the low energy
effective action for these phases. Each phase has two dual descriptions in
terms of antisymmetric tensors of different ranks, which are massless for the
Coulomb phase but massive for the Higgs and confinement phases. We illustrate
our prescription in detail for compact QED in 4D. Compact QED and $O(2)$ models
in 3D, as well as a periodic scalar field in 2D (strings on a circle), are also
discussed. In this last case we show how $T$-duality is maintained if one
considers both worldsheet instantons and their duals. We also unify various
approaches to the problem of the axion mass in 4D string models. Finally we
discuss possible implications of our results for non-perturbative issues in
string theory.
