In this talk, I describe some recent results on the Bogolubov-de Gennes equations. These equations give an equivalent formulation of the BCS theory of superconductivity.

I will explain how these equations emerge from the $N$-body Schrödinger equations and will discuss their general features and key physical classes of stationary solutions (normal, superconducting, vortex and vortex lattice states). I will describe results on existence of the normal, superconducting and vortex lattice states for non-zero magnetic fields and stability/instability of the normal states for large/small temperature or/and magnetic fields.

Then I will mention briefly the derivation of the macroscopic Ginzburg-Landau equations and describe some new results about these equations.

The talk is based on joint work with Li Chen and with V. Bach, S. Breteaux, Th. Chen and J. Fr\"{o}hlich and D. Chouchkov, N. Ercolani and S. Rayan.