We investigate the evolution of the particle size distribution of a coalescing particle field under different conditions in turbulence and the role of hydrodynamic interactions on the coalescing rate for near-contact motion of inertial droplets in quiescent flow. The primary motivation of this work is to understand the evolution of atmospheric clouds. The 10 – 50 microns size range in the cloud droplet growth evolution, often referred to as the ‘size gap’, is underpredicted by current microphysical models. There is growing consensus that turbulence plays a critical role in accelerating the cloud evolution. The first part of the study was performed using direct numerical simulation (DNS) of an evolving Eulerian fluid velocity field with Lagrangian particle tracking. We present parametric studies of the effects of critical variables on the particle size distribution of an initially monodisperse particle field as it collides and coalesces in turbulence without hydrodynamic interactions. We describe a Collision Optimized Detection Algorithm (CODA), embedded in our DNS turbulence code, to identify and enact particle coalescence in a manner that is optimized for a parallelized architecture. The effects of the particle sub-Kolmorogov size parameter, particle inertia, volume fraction and Reynolds number on the size distribution are systematically investigated. We find that the particle size distribution in turbulence: (i) has an exponential shape in terms of the particle Stokes number, a measure of particle inertia; (ii) has a very weak dependence on Reynolds number; (iii) broadens with decreasing particle Stokes number and decreasing size parameter; and (iv) transitions from an exponential to a power-law behavior at higher volume fractions. In the second part of the study, we investigate the importance of hydrodynamic interactions in near contact motion between the droplets in determining whether a collision leads to molecular contact and coalescence. As the droplets approach, the air in the gap must be squeezed out of the way, which leads to a resistance force that diverges with decreasing gap according to the continuum lubrication theory, preventing contact. At separations on the order of the mean free path of air, the continuum approximation breaks down, and the lubrication flow is described by a non-continuum model. Treating accurately the continuum nature of the gas at the far field and transitioning to a non-continuum model at gap separations comparable to the mean free path of the gas is therefore critical to capture the behavior leading up to contact and eventually coalescence. Building on previous work, which derived a uniformly valid expression for the resistivity to normal motion, we use a similar matched asymptotic expansion technique to derive the uniformly valid resistivity functions for tangential motions. In the third part of the study, we apply the complete set of resistivity functions to a trajectory analysis of droplet pairs settling in quiescent flow to investigate the collision efficiency as a function of the droplet size ratio, Knudsen number and Stokes number. In the near-contact motion, the collision efficiency increases with increasing pairwise droplet inertia and size ratio. It is observed to have a larger dependence on the Knudsen number for lower Stokes numbers, and the curves approach an asymptotic limit for Stokes numbers below 0.1. The collision efficiency for realistic cloud droplets at 50 microns peaks at 0.82. We conclude by discussing the implications of this work for cloud microphysics modeling and suggest next steps for future research.