Abstract : Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly and Thouless theory of the Quantum Hall Effect. It reduces to the critical almost Mathieu family, indexed by phase. We will present a complete proof of singular continuous spectrum for this family, for all phases, finishing a program with a long history, and based on a simple Fourier analysis and a new duality-type transform. We also present a result (with I. Krasovsky) that proves one half of the Thouless' one half conjecture from the early 80s: that Hausdorff dimension of the spectrum of Harper's operator is bounded by 1/2 for all irrational fluxes. If time permits, we will also discuss recent progress towards the Thouless Catalan conjecture (joint with I. Krasovsky and L. Konstantinov).