Grids are special families of tripotents in Jordan triple systems. This research monograph presents a theory of grids including their classification and coordinization of their cover. Among the applications given are — classification of simple Jordan triple systems covered by a grid, reproving and extending most of the known classification theorems for Jordan algebras and Jordan pairs — a Jordan—theoretic interpretation of the geometry of the 27 lines on a cubic surface — structure theories for Hilbert—triples and JBW*—triples, the Jordan analogues of Hilbert—triples and W*—algebras which describe certain symmetric Banach manifolds. The notes are essentially self—contained and independent of the structure theory of Jordan algebras and Jordan pairs. They can be read by anyone with a basic knowledge in algebraic geometry or functional analysis. The book is intended to serve both as a reference for researchers in Jordan theory and as an introductory textbook for newcomers to the subject.