A fundamental and arduous challenge in algebraic topology is to compute the stable homotopy groups of spheres. Chromatic homotopy theory decomposes this question into a "chromatic tower" of certain periodic families of generalized cohomologies. Originally invented to study complex cobordism classes, Morava K-theory and E-theory emerged as the prominent machinery central to the construction of these chromatic towers. This talk is an idiosyncratic attempt to introduce this field to anyone with a moderate background in algebraic topology. If time permits, I will also briefly discuss some recent advances in the computation of Morava E-theories and their homotopy fixed point spectra.