A natural orbital functional (NOF) satisfying the properties of the reduced density matrices is critically reviewed. Test calculations of the total and ionization energies, dipole moments, equilibrium geometries and harmonic vibrational frequencies of several molecules in the ground state demonstrate the reliability of the formalism. The extension of NOF theory to calculate correlation energies for periodic polymers is analyzed. The experimentally observed reduction of the Hartree-Fock lattice spacing by the inclusion of electron correlation is correctly predicted for trans-polyacetylene.