A general framework for integrable discrete systems on R, in particular, containing lattice soliton systems and their q-deformed analogs, is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, in terms of which one can define algebra of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures and their continuous limits are constructed. The inverse problem based on the deformation quantization scheme is considered
