The authors study density models for streamers viewed as nonlinear structures with multiple inner scales. They begin with a review of basic observations, experiments and microphysics and continue the article with a mathematical model consisting in reaction-drift-diffusion equations for charged particle densities coupled with the Poisson equations of electrostatics. They focus on the derivation and solution of moving boundary approximations for the density models. Recalling the fact that the negative streamers are linearly stable against branching they conjecture this for positive streamers too and identify open problems in the moving boundary approximations.