box splines are multivariate compactly supported piecewise polynomial functions, initially introduced in the 1980s to tackle problems of interpolation and approximation in several variables. it quickly transpired, through the work of dahmen, micchelli, de boor, ron, and others, and that box splines naturally give rise to a fascinating algebraic structure, with connections to such diverse areas as polynomial ideals, d-modules, hyperplane arrangements, tiling and enumeration of integer points in zonotopes, parking functions on graphs, and even partition functions of statistical mechanics. some of these connections are established only recently in joint work with ron and xu as well as in the work of de concini, procesi, vergne, ardila, postnikov, sturmfels and others. the speaker will try to give an overview of the area, with the focus on some exciting new developments.