This is the written exposition of a talk for a general mathematical audience, held by the author at the DMV-meeting. Complexity and degeneration of bilinear maps may be described by a single data structure: the asymptotic spectrum. The paper gives an introduction to this theory. As a red thread it uses \(\omega\), the exponent of matrix multiplication. The paper has been supplemented by footnotes containing more details and by an appendix, which is an invitation to application-oriented algebraists and geometers to be become acquainted with the topic. It contains short new proofs for some results in the main text, conjectures, and relations to other topics.