  Multiply-connected Calabi-Yau threefolds are of particular interest for both
string theorists and mathematicians. Recently it was pointed out that one of
the generic degenerations of these spaces (occurring at codimension one in
moduli space) is an isolated singularity which is a finite cyclic quotient of
the conifold; these were called hyperconifolds. It was also shown that if the
order of the quotient group is even, such singular varieties have projective
crepant resolutions, which are therefore smooth Calabi-Yau manifolds. The
resulting topological transitions were called hyperconifold transitions, and
change the fundamental group as well as the Hodge numbers. Here Batyrev's
construction of Calabi-Yau hypersurfaces in toric fourfolds is used to
demonstrate that certain compact examples containing the remaining
hyperconifolds - the Z_3 and Z_5 cases - also have Calabi-Yau resolutions. The
mirrors of the resulting transitions are studied and it is found, surprisingly,
that they are ordinary conifold transitions. These are the first examples of
conifold transitions with mirrors which are more exotic extremal transitions.
The new hyperconifold transitions are also used to construct a small number of
new Calabi-Yau manifolds, with small Hodge numbers and fundamental group Z_3 or
Z_5. Finally, it is demonstrated that a hyperconifold is a physically sensible
background in Type IIB string theory. In analogy to the conifold case,
non-perturbative dynamics smooth the physical moduli space, such that
hyperconifold transitions correspond to non-singular processes in the full
theory.
