This is an extensive and most readable survey of an algebraic approach to the foundation of the geometry of the universal Grassmann manifold, presented along the lines sketched by M. Sato and Y. Sato in RIMS Kokyuroku 439, 30-46 (1981; Zbl 0507.58029). This approach should be contrasted with the functional-analytical formulation by G. Segal and G. Wilson in Publ. Math., Inst. Hautes Etud. Sci. 61, 5-65 (1985; Zbl 0592.35112)] and by A. Pressley and G. Segal, Loop groups (1986; Zbl 0618.22011). The perhaps most important advantage is that one can develop a theory not only on the basis of real and complex numbers but also on, e.g. p-adic numbers.
The survey has the following main sections: finite-dimensional and infinite-dimensional Grassmann manifolds, actions of linear groups on universal Grassmann manifold, and multi-component theory.