Stable homotopy theory has its origins with Freudenthal, who observed that the homotopy groups of spheres exhibit stable behavior under suspension. Stable phenomena had appeared earlier, in examples like singular cohomology. But through later developments that used generalized cohomology theories to reduce deep geometrical problems to computations in stable homotopy (like K-theory, or Thom’s work on cobordism), the subject established its centrality. There was a need for a topological setting in which stable phenomena could be systematically studied.
In this talk, I will introduce the stable homotopy category from two complementary points of view. First, I will describe how attempts to stabilize the homotopy category of spaces naturally lead to spectra, using the sphere spectrum as a guiding example. Second, I will explain how spectra arise more importantly as representing objects for generalized cohomology theories, beginning with singular cohomology and K-theory.

I hope to place emphasis on building intuition from concrete examples rather than formal foundations. I will also briefly indicate why one is led to consider structured models of spectra (like S-modules) and symmetric monoidal categories of spectra—and, if time permits, say a few words about highly structured ring spectra and E∞-rings.