Over the past several decades, rapid progress has been made toward the accurate characterization and control of atoms, made possible largely by the development of narrow-linewidth lasers and techniques for trapping and cooling at ultracold temperatures. Extending this progress to molecules will have exciting implications for chemistry, condensed matter physics, and precision tests of physics beyond the Standard Model. These possibilities are all consequences of the richness of molecular structure, which is governed by physics substantially different from that characterizing atomic structure. This same richness of structure, however, increases the complexity of any molecular experiment manyfold over its atomic counterpart, magnifying the difficulty of everything from trapping and cooling to the comparison of theory with experiment.;This thesis describes work performed over the past six years to establish the state of the art in manipulation and quantum control of ultracold molecules. Our molecules are produced via photoassociation of ultracold strontium atoms followed by spontaneous decay to a stable ground state. We describe a thorough set of measurements characterizing the rovibrational structure of very weakly bound (and therefore very large) 88Sr2 molecules from several different perspectives, including determinations of binding energies; linear, quadratic, and higher order Zeeman shifts; transition strengths between bound states; and lifetimes of narrow subradiant states. The physical intuition gained in these experiments applies generally to weakly bound diatomic molecules, and suggests extensive applications in precision measurement and metrology. In addition, we present a detailed analysis of the thermally broadened spectroscopic lineshape of molecules in a non-magic optical lattice trap, showing how such lineshapes can be used to directly determine the temperature of atoms or molecules in situ, addressing a long-standing problem in ultracold physics. Finally, we discuss the measurement of photofragment angular distributions produced by photodissociation, leading to an exploration of quantum-state-resolved ultracold chemistry.
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