Computation of the stable homotopy groups of spheres is a long-standing open problem in algebraic topology. I will introduce chromatic homotopy theory, which uses localization of categories to split this problem into easier pieces, called chromatic levels, which can be understood using the theory of formal group laws. Each chromatic level is a symmetric monoidal category, and we can study its Picard group. I will describe methods that we can use to approach these Picard groups, and talk more specifically about current work at the second chromatic level.