Calabi-Yau varieties are complex, compact algebraic varieties with a trivial canonical class. The Calabi-Yau 1- and 2-folds are very well known, whereas it is not even known how many homotopy types of Calabi-Yau 3-folds there are nor whether this number is finite. Since they are used in superstring theory, physicists have created extensive databases of such constructions (containing several of their basic numerical invariants)---almost half a billion of them. Yet, within the last year, a generalization of these constructions has been discovered, which extends the tried and true methods of toric geometry. After a telegraphic survey of the by now standard constructions, I will discuss the novel generalizations and focus on a surprising phenomenon discovered amongst these, detected by the simplest of the Gromov-Witten invariants. Even for smooth models, this indicates a need for generalizing and refining classification efforts such as C.T.C. Wall's so as to include various invariants motivated by superstring theory.