We give an overview of the coupled Cahn-Hilliard and Navier-Stokes (CHNS) equations and show that they provide a natural theoretical framework for studying turbulence in binary-fluid mixtures. In particular, we demonstrate, by using recent studies that we have carried out, how the CHNS system can be used to study (a) the turbulence- induced suppression of phase separation in a symmetric binary-fluid mixture, (b) turbulence-induced multifractal dynamics in a droplet of the minority phase in a background of the majority phase, (c) the description of antibubbles, and (d) the mathematical exploration of possible finite-time singularities in solutions of the three-dimensional CHNS equations.