Calabi-Yau manifolds are compact, complex Kähler manifolds that have trivial first Chern classes (over R). In most cases, we assume that they have finite fundamental groups. By the conjecture of Calabi (1957) proved by Yau (1977; 1979), there exists on every Calabi-Yau manifold a Kähler metric with vanishing Ricci curvature.Currently, research on Calabi-Yau manifolds is a central focus in both mathematics and mathematical physics. It is partially propelled by the prominent role the Calabi-Yau threefolds play in superstring theories. While many beautiful properties of Calabi-Yau manifolds have been discovered, more questions have been raised and probed. The landscape of various constructions, theories, conjectures, and above all the fast pace of progress in this subject, have made the research of Calabi-Yau manifolds an extremely active research field both in mathematics and in mathematical physics.