The current status of the problem of the extremal mechanical properties (modulus of elasticity, strength, thermal stability) of fibres and filamentary crystals that I elaborated over more than 50 years is presented. Two new integral energy indexes of the structure of polymers are proposed and evaluated — energy density of interatomic and intermolecular interactions, which determine many of the properties of oriented polymers and are important for predicting the mechanical properties. In addition to the theoretical modulus of elasticity and the theoretical strength of polymers (like the strength at absolute zero according to H. Mark and G. M. Bartenev), two new, practically important indexes are proposed — the maximum attainable moduli of elasticity and maximum attainable strength (in the given temperature-time conditions) are proposed. Models are developed and comparative calculations of the theoretical and maximum attainable moduli of elasticity, maximum attainable strength, and maximum thermal stability are performed with different methods for a number of linear (aliphatic and aromatic) and laminar (including carbon) structures. A comparison is made with the maximum values of the mechanical and thermal properties of the fundamental types of fibres and needle crystals based on aliphatic and aromatic polymers and materials with covalent bonds of laminar and three-dimensional structures. Oriented materials with the maximum possible properties among all of the elements of periods 2 and 3 in the Mendeleev Periodic System and their compounds — hexagonal boron nitride, graphite, and the linear carbon polymer, β-carbyne, as a material with the highest modulus and strength possible in nature, are determined. New integral coefficients of completion of the structure of fibres as the ratios of the moduli of elasticity and strength attained to their maximum attainable values are proposed and evaluated. These coefficients characterize the level of the actually attained properties of oriented polymeric materials and consequently the perfection of the manufacturing technology. Ways of obtaining oriented polymeric materials with the maximum possible properties are discussed.