Abstract: Theoretical and experimental investigations of the structure, energy spectra and magnetic properties of pure organic high-spin low-dimensional (1-D and 2-D) polymers with possible magnetic ordering (ferro- or antifero-) are reviewed. The nature and the magnitude of the spin exchange interaction between the electrons within the half-filled band in two types of polymers with magnetic ordering are considered: (i) polymers with intramolecular magnetic interaction - polymers with a continual pi system of conjugation; (ii) a system with intermolecular magnetic interaction - molecular radical crystals in which the elementary units are weakly interacting pi subsystems. Many examples of high-spin systems with different constitution, topology, geometry and symmetry, and different character of the magnetic interaction between the electrons: direct, kinetic, indirect (superexchange) are considered. Also different types of polaronic ferromagnets are studied. A new concept (approach) to organic metals is discussed.