dence of the relaxation rate coefficients on the internal energy of molecules and their sensitivity to details of the interaction potential has not been carried out. Here, we give a brief account of recent quantum dynamics calcu- lations of vibrational and rotational energy transfer in atom - diatom collisions at cold and ultracold tempera- tures. We focus on a few representative systems to illus- trate the main features of energy transfer in non-reactive atom - molecule collisions at ultracold temperatures and we show how the corresponding rate coefficients are in- fluenced by rotational or vibrational excitation of the molecule. also experimental and theoretical evidence that chemical reactions involving heavy-atom tunneling of carbon [12] and fluorine [13] atoms can occur with significant rate coefficients at low temperatures. Photoassociation experiments involving alkali-metal systems have stimulated considerable interest in chem- ical reactivity in alkali-metal dimer - alkali-metal atom collisions at ultracold temperatures [14]. Rearrangement collisions in identical particle alkali-metal trimer systems occur without energy barrier and recent studies have demonstrated that chemical reactions in alkali-metal atom - alkali-metal dimer collisions may be very fast at ultracold temperatures. Unlike tunneling-dominated reactions, the limiting values of the rate coefficients for alkali-metal systems are less sensitive to vibrational excitation of the dimer. 2 A. Vibrational and rotational relaxation 1. Collisions at cold and ultracold temperatures In this chapter we give an overview of recent studies of ultracold atom - molecule collisions focusing on non- reactive and reactive systems and the effect of vibrational excitation of the molecule on the collisional outcome. We will discuss both tunneling-dominated and barrierless re- actions and examine recent efforts in extending these studies to ionic systems as well as molecule - molecule systems. We consider mostly the novel aspects of colli- sional dynamics of atom - diatom systems at cold and ul- tracold temperatures with illustrative results for specific systems. For more comprehensive discussion of cold and ultracold collisions including reactive and non-reactive processes and the effect of external fields we refer the reader to several review articles [6, 8, 13–15] that have appeared in the last few years. For details of the theoret- ical formalisms we refer to the chapters by Hutson and by Tscherbul and Krems. As discussed in the chapter by Hutson, at very low energies, scattering is dominated by s-waves and the scattering cross section can be expressed in terms of a single parameter called the scattering length. For single-channel scattering where only elastic scattering is possible, the scattering length is a real quantity and the magnitude of the cross section in the s-wave limit is given by σ = 4 πa 2 where a is the scattering length. For multichannel scattering, as in vibrationally or rotationally inelastic collisions of molecules, the scattering length is a complex number and it is denoted as a$_{vj}$ = α$_{vj}$ − iβ$_{vj}$ where v and j are, respectively, the initial vibrational and rotational quantum numbers of the molecule [10, 19]. The limiting value of the elastic cross section in the presence of inelastic scattering is given by σ el vj = 4 π | a$_{vj}$ | 2 = 4 π ( α 2 vj + β 2 $_{vj}$). The total inelastic quenching cross section from a given initial rovibrational level of the molecule is related to the imaginary part of the scattering length through the relation σ in vj = 4 πβ$_{vj}$/ k$_{vj}$ where k$_{vj}$ is the wave vector in the incident channel. The quenching rate coefficient becomes constant at ultralow temperatures and is given by k in vj = 4 π ℏ β$_{vj}$/µ at zero temperature where µ is the reduced mass of the collisional system. Thus, the rovibrational relaxation rate coefficients attain finite values for different initial vibrational and rotational levels of the molecule. The dependence of the rate coefficients on v and j has been an important issue in cold molecule research because exothermic vibrational and rotational relaxation collisions are a major pathway for trap loss in cooling and trapping experiments. Theoretical studies of ultracold molecules received in- tense interest after the success of ultracold atom pho- toassociation and buffer gas cooling experiments which demonstrated that a wide array of molecular systems with thermal and non-thermal vibrational energy dis- tributions can be created in ultracold traps. The col- lisional loss of trapped molecules is an important issue in these experiments. Photoassociation produces molecules in highly excited vibrational levels. Whether the excited molecules decay by vibrational quenching or through chemical reaction is an intriguing question. While an extensive literature exists on the collisional relaxation of vibrationally excited molecules at elevated tempera- tures not much was known on the magnitude of the re- laxation rate coefficients at temperatures lower than one Kelvin. Though a few earlier reports [16–18] on atom - diatom collisions in the Wigner threshold regime had been published, a detailed investigation of the depen- Initial studies of rotational and vibrational relaxation of atom - molecule systems at cold and ultracold tem- peratures have mostly focused on van der Waals systems such as He − H$_{2}$ [19–22], He − CO [23, 24], and He − O$_{2}$ [25]. Owing to the importance of some of these systems in as- trophysical environments, extensive calculations of low- temperature behavior of rate coefficients have been per- formed for collisions of H$_{2}$ [19] and CO [23, 24] with II. INELASTIC ATOM - MOLECULE COLLISIONS