Homotopy theory has long benefited from the ideas and constructions of algebraic geometry. The heart of the "chromatic approach" to understanding stable homotopy theory is the theory of 1-dimensional commutative formal groups, and work of Hopkins introduced the language of stacks to algebraic topology. Unfortunately, just as in the derived category of a ring, many of the constructions are not sufficiently rigid to be able to talk about limits and colimits; we have to form homotopical versions instead. Goerss, Hopkins, and Miller recognized that if we make our homotopy theory problem harder, then we can actually lift whole cloth the moduli stack of elliptic curves to homotopy theory. This was the start of spectral algebraic geometry. Lurie vastly generalized this, showing how we can carry out many familiar constructions in algebraic geometry in spectral algebraic geometry, and it is rapidly becoming a primary tool. In this talk, I'll give a gentle introduction to this circle of ideas, focusing on intuition and a concrete grounding of the problems.