Moscow-Petersburg seminar on low-dimensional mathematics March 7, 2003, 16:00, room 311 of PDMI N.N.Vasiliev Monomial orderings and Groebner walk. Groebner walk is a recent algorithm for the computation of Groebner bases of ideals in the ring of polynomials. This algorithm is connected to very interesting geometric and combinatorial objects in the space of all monomial orderings. In this report some of these constructions will be presented. Also the structure of the space of monomial orderings will be described. --------------------------- http://www.pdmi.ras.ru/~lowdimma