The buckling of thin plates is interesting from the point of view of materials science, because it provides a rather unique opportunity of exploring the predictions of elasticity theory in non-trivial conditions and allows in a specific example to disentangle new physical effects from effects that can be explained in the framework of the fundamental equations. I review these basic Föppl-von Karman equations for the elasticity of thin plates and then study the two limits of weak and strong nonlinear regimes for buckling.