We will briefly review the main concepts and fascinating physics background of one-dimensional analytic quasiperiodic operators and related cocycles. We then present the highlights of our joint papers with Lingrui Ge, Jiangong You, Qi Zhou, and Xin Zhao on a non-commutative generalization of the classical Jensen's formula, the explanation of the mysteries of Avila's global theory and various global spectral corollaries.