This paper is devoted to the description of our recent results on the spectral behavior of one-dimensional adiabatic quasi-periodic Schrödinger operators. The specific operator we study is a slow periodic perturbation of an incommensurate periodic Schrödinger operator, and we are interested in energies where the perturbation creates a strong interaction between two consecutive bands of the background periodic operator. We describe the location of the spectrum and its nature and discuss the various new resonance phenomena due to the interaction of the spectral bands of the unperturbed periodic operator.