  Some frustrated magnets and superconducting arrays possess unusual symmetries
that cause the free energy or other physics of a $D$-dimensional quantum or
classical problem to be that of a different problem in a reduced dimension
$d<D$. Examples in two spatial dimensions include the square-lattice $p+ip$
superconducting array, the Heisenberg antiferromagnet on the checkerboard
lattice (studied by a combination of 1/S expansion and numerical transfer
matrix), and the ring-exchange superconducting array. Physical consequences are
discussed both for ``weak'' dimensional reduction, which appears only in the
ground state degeneracy, and ``strong'' dimensional reduction, which applies
throughout the phase diagram. The ``strong'' dimensional reduction cases have
the full lattice symmetry and do not decouple into independent chains, but
their phase diagrams, self-dualities, and correlation functions indicate a
reduced effective dimensionality. We find a general phase diagram for quantum
dimensional reduction models in two quantum dimensions with $N$-fold
anisotropy, and obtain the Kosterlitz-Thouless-like phase transition as a
deconfinement of dipoles of 3D solitons.
