Groups with root subgroups arise naturally in the study of Chevalley groups over rings, algebraic groups and buildings. The notion of a root grading provides a common, purely combinatorial, framework to capture these examples. The author carefully develops the theory assuming only a minimal background in algebra. This culminates in the main result that every root graded group of rank at least 3 is coordinatised by an algebraic structure over a ring in such a way that a variation of the Chevalley commutator formula is satisfied. This result can be regarded as a generalisation of Jacques Tits' classification of thick irreducible spherical buildings of rank at least 3 to the setting of non-division algebraic structures. It is proven in a characteristic-free way using a powerful new computational method: the blueprint technique. This comprehensive and self-contained treatment is ideal for graduate students and researchers in algebra, group theory, and building theory.