This talk will explain the philosophy behind web bases (like the double ladders basis) which are derived from branching rules. One is interested in studying morphisms between tensor products of generating representations. A branching graph is a generalization of the Young graph, with vertices associated to simple representations, and an edge whenever one simple is a summand of another simple tensored with a generating representation. These graphs are usually infinite. There is a tautological argument that one can construct bases for morphism spaces associated to paths in the branching graph, given data associated to each edge. We explain what a lifting of this data is, and how liftings give rise to web bases. We also explain the concept of branching patterns, which allows one to construct liftings for infinitely many edges (all of them) from a finite amount of data. Time permitting, we discuss the problem of clasp computation.