  While Calabi-Yau hypersurfaces in toric ambient spaces provide a huge number
of examples, theoretical considerations as well as applications to string
phenomenology often suggest a broader perspective. With even the question of
finiteness of diffeomorphism types of CY 3-folds unsettled, an important idea
is Reid's conjecture that the moduli spaces are connected by certain singular
transitions. We summarize the results of our recent construction of a large
class of new CY spaces with small Picard numbers and of their mirrors via
conifold transitions and discuss the benefits of other approaches to
interesting locations in the web that have been or should be pursued.
