The nonlinear curl-curl problems have recently arisen in the search for exact propagation of electromagnetic waves in non-linear media modelled with Maxwell equations. The quintic effect leads to the following critical curl-curl problem: $\nabla \times (\nabla \times u) = |u|^4 u$, where $u : R^3 \to R^3$ is the profile of the time-harmonic electric field. Ground states solutions of the problem are related with the optimizers of a Sobolev-type inequality involving the curl operator in $R^3$. We show that there is a ground state solution and infinitely many bound state solutions. Some symmetric properties of the problem and extensions to the $p$-curl-curl equation in the critical case with applications to zero modes of the Dirac equation will be also discussed.