Thermodynamic fluctuations are significant at microscopic scales even when hydrodynamic transport models (i.e., Navier-Stokes equations) are still accurate; a well-known example is Rayleigh scattering, which makes the sky blue. Interesting phenomena also appear in non-equilibrium systems, such as the enhancement of diffusion during mixing due to the correlation of velocity and concentration fluctuations. Direct Simulation Monte Carlo (DSMC) simulations are useful in the study of hydrodynamic fluctuations due to their computational efficiency and ability to model molecular detail, such as internal energy and chemical reactions. More recently, finite volume schemes based on the fluctuating hydrodynamic equations of Landau and Lifshitz have been formulated and validated by comparisons with DSMC simulations. This paper discusses some of the relevant numerical issues and physical effects investigated using DSMC and stochastic Navier-Stokes simulations. This paper also presents the multi-component fluctuating hydrodynamic equations, including chemical reactions, and illustrates their numerical solutions in the study of Turing patterns. We find that behind a propagating reaction front, labyrinth patterns are produced due to the coupling of reactions and species diffusion. In general, fluctuations accelerate the propagation speed of the leading front but differences are observed in the Turing patterns depending on the origin of the fluctuations (stochastic hydrodynamic fluxes versus Langevin chemistry).