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A range of ultracold molecular problems were solved, leading to new opportunities for advanced material development and quantum simulators. These molecules include (1) the already quantum degenerate bialkali singlet sigma case potassium-rubidium (KRb) and related molecules; (2) opto-electrically trapped symmetric top molecules soon to reach quantum degeneracy and common in nature, like methyl fluoride (CH3F); and (3) the recently evaporatively cooled hydroxyl free radical (OH). Research achievements on this grant cycle include (A) a fundamental understanding of how strongly-interacting fermions pair to make molecules in the presence of a lattice; (B) experimental guidance and specific numerical predictions for five bi-alkali molecules being studied in labs around the world today, including potassium-rubidium; (C) a correction of phase diagrams for dipolar gases necessary to understand experimental measurements and build accurate quantum simulators; (C) a whole new area of quantum many body physics opened up in symmetric top molecules, involving e.g. exotic tunable XYZ magnetism and quantum liquid crystals; and (D) new high precision predictions for the spectra of the hydroxyl free radical absolutely necessary to reach quantum degeneracy. This work resulted in eight publications and papers under review in peer-reviewed journals. Synergistic activities included (i) organizing an extended program at the Kavli Institute of Theoretical Physics and Kavli symposia for the National Academy of Sciences; (ii) editing a special issue of the New Journal of Physics bringing ultracold molecule researchers from around the world together with a planned collection of over sixty publications; (iii) an edited series on cold atoms and molecules; and (iv) thirty-two invited seminars, sixteen directly on the topic of this grant.