We will give a brief introduction to the spectral theory of  ergodic operators. Then we will discuss several remarkable spectral  phenomena present in the class of quasiperiodic operators and  illustrate using the almost Mathieu (aka Harper's) operator - a model  behind the Hofstadter's butterfly and Thouless theory of the Quantum  Hall Effect. We will discuss the fascinating history of this model,  that is now heavily studied in physics, and then will describe several  recent results that resolve some long-standing conjectures.