Calabi–Yau threefolds: Age of Discovery?

I'll give a brief overview of the so far discovered borders in the geography of Calabi–Yau threefolds. Mostly I'll be concerned with two (conjecturally almost symmetric) borders: Calabi–Yau threefolds with Picard number one (and also with the related question of Fano fourfolds with Picard number one), and Calabi–Yau threefolds with one-dimensional moduli space (also rigid Calabi–Yau threefolds, that violate mirror symmetry).
I know about a hundred examples, including some hypothetical, and about fifty examples that I am confident in.
There are less than ten various constructions, that I’ll try to review.
Time permits, I’ll tell some new constructions that produce about a dozen new examples (or realize some of the old hypotheses).