A pattern is a list of positions in an n×n real matrix. A matrix completion problem for the class of Π-matrices asks whether every partial Π-matrix whose specified entries are exactly the positions of the pattern can be completed to a Π-matrix. We survey the current state of research on Π-matrix completion problems for many subclasses Π of P0-matrices, including positive definite matrices, M-matrices, inverse M-matrices, P-matrices, and matrices defined by various sign symmetry and positivity conditions on P0- and P-matrices. Graph theoretic techniques used to study completion problems are discussed. Several new results are also presented, including the solution to the M0-matrix completion problem and the sign symmetric P0-matrix completion problem.