The conformation of a polypeptide or protein chain may be specified by stating the orientations of the two linked peptide residues at each alpha carbon atom in the chain, namely the two dihedral angles varphi, varphi' about the single bonds N-alphaC and alphaC-C' from a defined standard conformation. By using certain criteria of minimum contact distances between the various atoms, the allowed anges of (varphi, varphi') have been worked out for three values of the angle N-alphaC-C' (tau), namely 105, 110, and 115 degrees for non-glycyl, and 110 and 115 degrees for glycyl residues. The theory is compared with all the available crystallographic data (up to early 1965) on simple (di- and tri-) peptides, cyclic peptides, polypeptide and protein structures, and the observed data fully support the conclusions from theory. The effect of the gamma carbon atom, in its three possible positions, is also discussed, and is found to alter the outer limits of the allowed region of (varphi, varphi') only slightly. The paper contains exhaustive references to the published data on these structures, using x-ray diffraction.