Low-rank matrix completion is a well-studied problem, for which a rich theory exists, mainly from the optimization and statistical points of view. However, the majority of the theoretical analysis in the literature assumes random observation patterns. On the other hand, the pattern of observed entries depends on the nature of each application, and is usually not random. This leads to the following fundamental question: What are the uniquely completable patterns at a certain rank r? This is a natural question of algebraic geometry that we will undertake in this talk (no prior experience with algebraic geometry will be assumed).