  This is a short review of recent constructions of new Calabi-Yau threefolds
with small Hodge numbers and/or non-trivial fundamental group, which are of
particular interest for model-building in the context of heterotic string
theory. The two main tools are topological transitions and taking quotients by
actions of discrete groups. Both of these techniques can produce new manifolds
from existing ones, and they have been used to bring many new specimens to the
previously sparse corner of the Calabi-Yau zoo where both Hodge numbers are
small. Two new manifolds are also obtained here from hyperconifold transitions,
including the first example with fundamental group S3, the smallest non-Abelian
group.
