Collisions between atoms, ions, and molecules play a fundamental role in a number of astrophysical contexts. In this work, we perform quantum mechanical close-coupling scattering calculations for a variety of collision systems and discuss their implications for our understanding of various astrophysical processes. We begin with a study of rovibrational de-excitation of HD in collisions with He. Along with H2, HD has been found to play an important role in the cooling of the primordial gas in the formation of the first stars and galaxies, and the rate of this cooling requires a knowledge of collision rate coefficients with common neutrals such as H and He. In this study we perform cross section calculations for the He-HD collision system over a range of collision energies and for initial rovibrational states of j=0 and 1 for v=0 to 17. We report rate coefficients for all Delta v=0, -1, and -2 transitions and compare them to previous calculations. Next we examine the effect of theoretically varying the collision-system reduced mass in collisions of He with vibrationally excited molecular hydrogen. Complex scattering lengths and vibrational quenching cross sections, and a low-energy scattering resonance are studied as a function of the collision system reduced mass. Experimental observations of these phenomena in the ultracold regimes for collisions of He with H2, HD, HT, and DT should be feasible in the near future. Finally, we perform electron-capture cross section calculations for the collision systems O7+ + H and C5+ + H using the quantum mechanical molecular orbital close-coupling method. Charge exchange between highly-charged solar wind ions and neutral interstellar hydrogen has been found to be a significant contributor to the heliospheric component of the soft x-ray background, as the highly excited resultant ions emit x-rays in the electron's cascade to the ground state. Calculations are performed over a range of collision energies for all important n-, l-, and S-resolved states. We compare our results to new atomic orbital close-coupling, classical trajectory Monte Carlo, and experimental merged-beam results.