  In this talk, we present three examples of new non-trivial vacuum structures
that can occur in supersymmetric field theories, along with explicit models in
which they arise. The first vacuum structure is one in which supersymmetry is
broken at tree-level in a perturbative theory that also contains a
supersymmetry-preserving ground state. Models realizing this structure are
uniquely characterized by the fact that no flat directions appear in the
classical potential, all vacua appear at finite distances in field space, and
no non-perturbative physics is required for vacuum stability. The second
non-trivial vacuum structure we discuss consists of large (and even infinite)
towers of metastable vacua, and we show that models which give rise to such
vacuum towers exhibit a rich set of instanton-induced vacuum tunneling
dynamics. Finally, our third new non-trivial vacuum structure consists of an
infinite number of degenerate vacua; this leads to a Bloch-wave ground state
and a vacuum "band" structure. Models with such characteristics therefore
experience time-dependent vacuum oscillations. Needless to say, these novel
vacuum structures lead to many new potential applications for supersymmetric
field theories, ranging from the cosmological-constant problem to the string
landscape, supersymmetry breaking, and Z' phenomenology.
