Algorithms for stable marriage and related matching problems typically assume that full preference information is available. While the Gale-Shapley algorithm can be viewed as a means of eliciting preferences incrementally, it does not prescribe a general means for matching with incomplete information, nor is it designed to minimize elicitation. We propose the use of maximum regret to measure the (inverse) degree of stability of a matching with partial preferences; minimax regret to find matchings that are maximally stable in the presence of partial preferences; and heuristic elicitation schemes that use max regret to determine relevant preference queries. We show that several of our schemes find stable matchings while eliciting considerably less preference information than Gale-Shapley and are much more appropriate in settings where approximate stability is viable.