In algebra, localization and completion at prime ideals are powerful tools. Chromatic homotopy studies these concepts in the stable homotopy category. In these lectures, I will give an introduction to this field.

Our first goal will be to understand the primes of the stable homotopy category. To this end, I will start by introducing Bousfield localization and reviewing examples of this construction. We will spend quite a bit of time understanding localization at $K$-theory. Then, we will see the generalization of these ideas to higher chromatic heights: we will get to know the Morava $K$-theories and understand in what sense they are the primes of stable homotopy. A key player in our study of localization at $K$-theory will have been the telescopic localization of a finite spectrum at a self-map. We will introduce these at higher heights as well. At this point, I will be able to state the telescope conjecture. With the goal of understanding Morava $K$-theoretic localization, we will then take a detour and discuss deformation theory, eventually formulating the statement that Morava $E$-theory is a Galois extension of the $K(n)$-local sphere which can be thought of as its algebraic closure. Depending on how much time remains, we will dive into newer topics: first, we will discuss ambidexterity and intermediate Galois extensions of the $K(n)$-local sphere. This will allow me to explain the refinement of the Telescope Conjecture which, ultimately, led to its demise.