In three complex dimensions, classification of the possible Calabi-Yau manifolds is an open problem, although Yau suspects that there is a finite number of families (albeit a much bigger number than his estimate from 20 years ago).
For every positive integer n, the zero set of a non-singular homogeneous degree n+2 polynomial in the homogeneous coordinates of the complex projective space CPn+1 is a compact Calabi-Yau n-fold.