In this chapter we have given an overview of recent theoretical studies of atom - molecule and molecule - molecule collisions at cold and ultracold temperatures. Though such systems have been extensively studied at higher collision energies over the last few decades, the new experimental breakthroughs in creating dense samples of cold and ultracold molecules have provided unprecedented opportunities to explore elastic, inelastic, and reactive collisions at temperatures close to absolute zero. These studies have revealed unique aspects of molecular collisions and energy transfer mechanisms that are otherwise not evident in thermal energy collisions. 

The long duration of collisions combined with large de Broglie wavelengths at cold and ultracold temperatures leads to interesting quantum effects. Calculations have shown that reactions with insurmountable energy barriers may still occur at temperatures close to absolute zero, and in certain cases, with appreciable rate coeffi- cients. Such tunneling dominated reactions have been the topic of many recent investigations and may soon be amenable to experimental investigation in the cold and ultracold regime. That the rates of these reactions can be enhanced by vibrational excitation of the molecule is an interesting scenario for experimental studies of ultracold chemical reactions. 

The studies of alkali-metal homonuclear and het- eronuclear trimer systems have stimulated considerable experimental interest in investigating chemical reactivity at ultracold temperatures. The challenge for theory is to describe collisions involving highly vibrationally excited molecules. Recent theoretical calculations have indicated that the three-body interaction potential can be neglected for highly vibrationally excited molecules 

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## V. SUMMARY AND OUTLOOK 

offering significant savings in computational effort. Even so, heavier alkali-metal trimer systems pose a daunting computational challenge. 

The quantitative description of ultracold molecule - molecule collisions is another challenging topic. The recent progress on the H$_{2}$ − H$_{2}$ system will be difficult to implement for heavier systems due to the large number of ro-vibrational levels of the molecules. The study of H$_{2}$ − H$_{2}$ collisions has shown that, for certain combinations of ro-vibrational levels, the energy transfer may occur to specific final ro-vibrational states. In such cases, the calculations can use a much smaller basis set without compromising the accuracy. 

Currently there is substantial interest in controlling the collisional outcome using external electric and magnetic fields. While the idea of coherent control of molecular collisions and chemical reactivity has existed for a long time and some important progress has been achieved, the possibility of creating coherent and dense samples of molecules in specific quantum states has given further impetus to the field of controlled chemistry. We expect that the coming years will see a far greater activity in this direction driven by cold and ultracold molecules and also by the possibility of controlling chemical reactivity using external electric and magnetic fields. Electronically non-adiabatic effects in ultracold collisions is a largely unexplored area, which can be expected to attract much attention. 

Acknowledgments: This work was supported by NSF grants # PHY-0555565 (N.B.), AST-0607524 (N.B.), and by the Chemical Science, Geoscience and Bioscience Di- vision of the Office of Basic Energy Science, Office of Science, U.S. Department of Energy (A.D.). 

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