  Calabi-Yau spaces, or Kahler spaces admitting zero Ricci curvature, have
played a pivotal role in theoretical physics and pure mathematics for the last
half-century. In physics, they constituted the first and natural solution to
compactification of superstring theory to our 4-dimensional universe, primarily
due to one of their equivalent definitions being the admittance of covariantly
constant spinors. Since the mid-1980s, physicists and mathematicians have
joined forces in creating explicit examples of Calabi-Yau spaces, compiling
databases of formidable size, including the complete intersecion (CICY)
dataset, the weighted hypersurfaces dataset, the elliptic-fibration dataset,
the Kreuzer-Skarke toric hypersurface dataset, generalized CICYs etc., totaling
at least on the order of 10^10 manifolds. These all contribute to the vast
string landscape, the multitude of possible vacuum solutions to string
compactification. More recently, this collaboration has been enriched by
computer science and data science, the former, in bench-marking the complexity
of the algorithms in computing geometric quantities and the latter, in applying
techniques such as machine-learning in extracting unexpected information. These
endeavours, inspired by the physics of the string landscape, have rendered the
investigation of Calabi-Yau spaces one of the most exciting and
inter-disciplinary fields.
  Invited contribution to the Oxford Research Encyclopedia of Physics,
B.~Foster Ed., OUP, 2020
