Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings (including this one)[15, 26, 41, 66]; three workshop proceedings [33, 47, 77]; five special journal issues [5, 88, 24, 40, 67]; more than ten surveys [2, 7, 27, 28, 44, 56, 57, 76, 82, 81]; one edited collection of papers [1]; four monographs [3, 12, 55, 65]; and seven books (four of them still in progress)[8, 9, 35, 54, 60, 75, 84]. To encourage and stimulate continued progress in this area, we have collected (with the help of colleagues) a number of problems that appear to us to be of interest and regarding which we do not know the answer. Questions on rewriting and other equational paradigms have been included; many have not aged sufficiently to be accorded the appellation" open problem". We have limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementations. Previous lists of questions in this area include one distributed by Leo Marcus and one of us (Dershowitz) at the Sixth International Conference on Automated Deduction (New York, 1982), the questions posed in a set of lecture notes on" Term Rewriting Systems" by one of us (Klop) for a seminar on reduction machines (Ustica, 1985), another list by one of us (Jouannaud) in the Bulletin of the European Association for Theoretical Computer Science (Number 31, 1987), and electronic postings to the distribution list (rewri~ ing~ crin. cr�n. fr) maintained by Pierre Lescarme. We use primarily terminology and notation of [27].