The combinatorial spider is a diagrammatic category that encodes quantum sl(n) representations, and was formalized by Kuperberg. Webs are certain directed planar graphs (with edge-weights), corresponding to the morphisms in this category, and endowed with skein-type relations that indicate algebraic equivalences. Webs are well-understood in the case n=2, when they are essentially noncrossing matchings (or Temperley-Lieb diagrams), and in the substantially more complicated case n=3.

In this talk, we sketch some of the historical evolution of webs, including work of Kuperberg, Khovanov, Fontaine, and Cautis-Kamnitzer-Morrison, as well as the convergence with a collection of combinatorial ideas about plabic graphs from Postnikov, Fomin-Pylyavskyy, Fraser-Lam-Le, and others. We also describe a new approach, joint with Heather M. Russell, that uses a set of colored paths called \emph{strands} to give a global construction for webs, via graph-theoretic and combinatorial notions generalized from smaller dimensions. Time permitting, we'll also allude to connections to algebraic geometry.