Rotationally and vibrationally inelastic collision dynamics of several small molecules are investigated through ab initio calculations of potential energy surfaces (PESs) and time-independent close-coupling scattering calculations. The scattering resonances in the collision energy dependent rotationally inelastic cross sections of OH in collisions with He and Ne, and NH3 in collisions with H2 were computed and analyzed. Both shape and Feshbach resonances were identified and the prospects for experimentally observing scattering resonances using Stark decelerated beams of OH radicals were discussed. A new PES for the interaction between CH3 with different umbrella displacements and a He atom were computed and the collisional vibrational relaxation of the ν2 mode of CH3 were studied. The vibrational relaxation rate constant was found to be two orders of magnitude smaller than the pure-rotational relaxation between two lower levels. Differential cross sections for the rotationally inelastic scattering of CH3 and CD3 with He, Ar, and H2 were computed and compared with results of velocity map imaging experiments conducted by Orr-Ewing and coworkers. In general, good agreement was found between theory and experiment, confirming the accuracy of our theoretical approach. Also, new sets of PESs describing the interaction between OH and H2 were computed, and bound-state calculations and scattering calculations were performed for this system. The computed dissociation energy of OH--\emph{ortho}-H2 complex and state-to-state cross sections of OH in collisions with H2 are in excellent agreement with earlier experimental results
