Title : The spectral theory of adiabatic quasi-periodic operators Abstract : The talk is devoted to recent results obtained in collaboration with A. Fedotov (St Petersburg (Russia)) on the spectral properties of quasi-periodic Schrodinger operators on the real line. The potential under consideration is the sum of a periodic part and of an adiabatic quasi-periodic "perturbation". We study the locus and the nature of the spectrum of this operator. Therefore, we introduce the monodromy matrix that will serve as a local model for our equation. This local model enables us to describe the locus of the spectrum in terms of quantization conditions associated to a natural adiabatic Hamiltonian. The nature of the spectrum is described in terms of phase space tunneling. This enables us to study the occurrence of Anderson transitions for the class of operators under consideration.