Abstract: By the Brown representability theorem, cohomology theories in algebraic topology are represented by spectra. While it is easy to define an additive structure on these spectrum objects, describing a product structure that reflects the underlying product on cohomology is a surprisingly difficult task. Starting from the classical notion of cohomology theories and sequential spectrum, I will describe how this problem is solved using orthogonal spectra, and time permitting introduce $E_\infty$ ring spectra. This talk should be accessible to grad students who know basic algebraic topology, specifically, Brown representability.