A Calabi-Yau manifold is a compact Kahler manifold with trivial canonical bundle. Yau’s solution to the Calabi conjecture yields canonical Ricci-flat Kahler metrics (Calabi-Yau metric) on such a manifold, and this has deep applications in many areas of mathematics. It is a longstanding question to understand how the Ricci-flat metrics develop singularities when the complex structure degenerates. An especially intriguing phenomenon is that these metrics can collapse to lower dimensions and exhibit very non-algebraic features, and it is challenging  to describe the corresponding geometric behavior.  In this talk I will review the status of this problem and explain some recent progress.