A mixing layer is a common model used to study the noise generation and mixing characteristics of the near-nozzle region of jets. This work presents three separate but related studies that investigate sound generation and active control for noise mitigation and mixing enhancement of such mixing layers. High-fidelity direct numerical simulations of temporal and spatial mixing layers are used for this in two and three dimensions. The first study investigates the role of turbulence scales in generating the radiated far-field sound from temporally-developing, Mach 0.9 mixing layers. To do this, four mixing layers were simulated, starting from the same initial conditions but with Reynolds numbers that varied by a factor of twelve. Above a momentum thickness Reynolds number of 300, all the mixing layers radiate over 85 percent of the acoustic energy of the apparently asymptotically high-Reynolds-number value we are able to compute. Wavenumber spectra of turbulence energy and pressure show the expected Reynolds number dependence: the two highest Reynolds number simulations show evidence of an inertial range and Kolmogorov scaling at the highest wavenumbers. Far-field pressure spectra all decay much more rapidly with wavenumber than the corresponding near-field spectra and show significantly less sensitivity to Reynolds number. Low wavenumbers account for nearly all of the radiated acoustic energy. Implications of these results for jet noise large-eddy simulations are discussed. The second study uses direct numerical simulations of Mach 1.3 mixing layers to characterize the physical mechanisms of flow actuation by localized arc-filament plasma actuators. A validated numerical model of the actuator is devised and placed, as in corresponding experiments, in a cavity in the nozzle near its exit. A rapid Joule heating caused by the plasma is thought to be the root mechanism of flow actuation based upon experimental observation. Simulations show that in the confined space of the cavity, the actuator creates a rapid flow expansion, which transfers fluid mass upward and outward creating a synthetic-jet-like perturbation to the boundary layer. The actuation promotes vortex creation much closer to the nozzle than the baseline flow without actuation, increases the layer growth rate, and organizes the large flow structures. Placing the actuator in a cavity of half the original width increases the velocities responsible for the jet-like boundary layer perturbation and downstream mixing layer growth rate. An actuator model designed to produce the same pressure response without the rapid heating provides similar control authority. The final study implements an automatic optimization procedure based on the adjoint of the perturbed and linearized flow equations. An algorithm is formulated to provide optimized control actuation for noise reduction and mixing enhancement objectives. The method is demonstrated to be successful on several model problems in two and three dimensions, in cases both with an explicitly represented "splitter" plate and cases where an appropriate inflow condition is imposed in its place. Cost functionals for noise reduction and mixing enhancement based on cross-stream velocity and pressure are formulated. Two-dimensional mixing layers with near-wall control are presented with velocity- and pressure-based spreading enhancement cost functionals. Both controls are able to maximize their respective cost functionals by over 50% and increase mixing layer thickness by 10-15% over the optimization time horizon. A three-dimensional, turbulent (spatially-developing) mixing layer is simulated and optimized with a noise reduction cost functional. The control successfully reduces the noise on a target plane below the mixing layer by 28% after 4 line search iterations of the optimization scheme.