The counterparts of the Urysohn universal space in category of metric spaces and the Gurarii space in category of Banach spaces are constructed for separable valued Abelian groups of fixed (finite) exponents (and for valued groups of similar type) and their uniqueness is established. Geometry of these groups, denoted by G_r(N), is investigated and it is shown that each of G_r(N)'s is homeomorphic to the Hilbert space l^2. Those of G_r(N)'s which are Urysohn as metric spaces are recognized. `Linear-like' structures on G_r(N) are studied and it is proved that every separable metrizable topological vector space may be enlarged to G_r(0) with a `linear-like' structure which extends the linear structure of the given space.

60 pages