  Many complex systems exhibit hydrodynamic (or macroscopic) behavior at large
scales characterized by few variables such as the particle number density,
temperature and pressure obeying a set of hydrodynamic (or macroscopic)
equations. Does the hydrodynamic description exist also for waves in complex
open media? This is a long-standing fundamental problem in studies on wave
localization. Practically, if it does exist, owing to its simplicity
macroscopic equations can be mastered far more easily than sophisticated
microscopic theories of wave localization especially for experimentalists. The
purposes of the present paper are two-fold. On the one hand, it is devoted to a
review of substantial recent progress in this subject. We show that in random
open media the wave energy density obeys a highly unconventional macroscopic
diffusion equation at scales much larger than the elastic mean free path. The
diffusion coefficient is inhomogeneous in space; most strikingly, as a function
of the distance to the interface, it displays novel single parameter scaling
which captures the impact of rare high-transmission states that dominate
long-time transport of localized waves. We review aspects of this novel
macroscopic diffusive phenomenon. On the other hand, it is devoted to a review
of the supersymmetric field theory of light localization in open media. In
particular, we review its application in establishing a microscopic theory of
the aforementioned unconventional diffusive phenomenon.
