A partial matrix is one in which some entries are specified, while
 the remaining unspecified entries are free to be chosen from an
agreed upon set. A completion of a partial matrix is a choice of
 values from the unspecified entries, resulting in a conventional
 matrix. A matrix completion problem asks which partial matrices have
 completions with a property of interest, say positive definite, a
 certain rank, etc. We discuss work of the last several years on
 matrix completion problem, including the important role of the
 pattern of the specified entries.