Fluid mixing that is induced by hydrodynamic instability is ubiquitous in nature; the material interface between two fluids when perturbed even slightly, changes shape under the influence of hydrodynamic forces, and an additional zone called the mixing layer where the two fluids mix, develops and grows in size. This dissertation reports a study on the role of thermal fluctuations in fluid mixing at the interface separating two perfectly miscible fluids of different densities. Mixing under the influence of two types of instabilities is studied; the Rayleigh-Taylor (RTI) and Richtmyer-Meshkov (RMI) instabilities. The study was conducted using numerical simulations after verification of the simulation methodology. Specifically, fluctuating hydrodynamic simulations were used; the fluctuating compressible Navier-Stokes equations were the physical model of the system, and they were solved using numerical methods that were developed and implemented in-house. Our results indicate that thermal fluctuations can trigger the onset of RTI at an initially unperturbed fluid-fluid interface, which subsequently leads to mixing of multi-mode character. In addition we find that for both RMI and RTI, whether or not thermal fluctuations quantitatively affect the mixing behavior, depends on the magnitude of the dimensionless Boltzmann number of the hydrodynamic system in question, and not solely on its size. When the Boltzmann number is much smaller than unity, the quantitative effect of thermal fluctuations on the mixing behavior is negligible. Under this circumstance, we show that mixing behavior is the average of the outcome from several stochastic instances, with the ensemble of stochastic instances providing the bounds on mixing-related metrics such as the mixing width. Most macroscopic hydrodynamic systems fall in this category. However, when the system is such that the Boltzmann number is of order unity, we show that thermal fluctuations can significantly affect the mixing behavior; the ensemble-averaged solution shows a departure from the deterministic solution. We conclude that for such systems, it is important to account for thermal fluctuations in order to correctly capture their physical behavior.