Splines in geometry and topology

•Many varieties X have a graph GXGX that describes their essential geometric features.•The splines S∞0(GX) form the equivariant cohomology ring and encode key data about X.•We describe geometric tools to analyze splines that differ from classical tools.•We give an algebraic and combinatorial generalization called generalized splines.

This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.