In the field of computational fluid dynamics, lattice gas and lattice Boltzmann methods are powerful simulation methods derived from kinetic theory. These methods are renowned for their simplicity of implementation and computational speed. In recent years, lattice Boltzmann has risen in popularity for modeling hydrodynamic flows, diffusion, and more. However, a limitation of these methods is the lack of fluctuations due to the continuous nature of the model. Fluctuations arise from the discreteness found in nature, so including fluctuations presents difficulties. This dissertation explores new and novel ways of improving lattice Boltzmann and lattice gas methods. First, we present a new derivation for a fluctuating lattice Boltzmann method in a diffusive system. Fluctuations are absent lattice Boltzmann since they were derived as a Boltzmann average of discrete lattice gases. This lattice Boltzmann method is exact and includes density dependent noise which models fluctuations to high accuracy. Second, we extend diffusive lattice Boltzmann methods to apply to physical systems for diffusion through barrier coatings. We found that these models were able to reproduce the behavior from previous experiments and provided a simple tool for analyzing such systems. Higher order corrections to lattice Boltzmann methods are explored for extending the range for successful lattice Boltzmann implementations. Recently, the implementation of an integer lattice gas with a Monte Carlo collision operator by Blommel et al. provided a template for incorporating fluctuations through the discrete nature of lattice gases. A sampling collision operator for integer lattice gases by Seekins et al. was able to reproduce the fluctuating diffusion equation in the Boltzmann limit similar to the diffusive fluctuating lattice Boltzmann. However, lattice gases have a more limited range of transport coefficients than lattice Boltzmann methods, since lattice Boltzmann collisions are deterministic and allow for the implementation of over-relaxation and lattice gas collisions are probabilistic and overrelaxation in a lattice gas requires a probability greater than 1. The final section of this dissertation presents a simple method for including overrelaxation into an integer lattice gas using the sampling collision operator. It will be shown that this is possible through a permutation of occupation numbers.