  Binary fluid mixtures are examples of complex fluids whose microstructure and
flow are strongly coupled. For pairs of simple fluids, the microstructure
consists of droplets or bicontinuous demixed domains and the physics is
controlled by the interfaces between these domains. At continuum level, the
structure is defined by a composition field whose gradients which are steep
near interfaces drive its diffusive current. These gradients also cause
thermodynamic stresses which can drive fluid flow. Fluid flow in turn advects
the composition field, while thermal noise creates additional random fluxes
that allow the system to explore its configuration space and move towards the
Boltzmann distribution. This article introduces continuum models of binary
fluids, first covering some well-studied areas such as the thermodynamics and
kinetics of phase separation, and emulsion stability. We then address cases
where one of the fluid components has anisotropic structure at mesoscopic
scales creating nematic (or polar) liquid-crystalline order; this can be
described through an additional tensor (or vector) order parameter field. We
conclude by outlining a thriving area of current research, namely active
emulsions, in which one of the binary components consists of living or
synthetic material that is continuously converting chemical energy into
mechanical work.

Soft Condensed Matter