The biennial review of atomic weight, A-r(E), determinations and other cognate data have resulted in the following changes in recommended values (1979 values in parentheses): Hydrogen 1.00794 [plus or minus] 7 (1.0079 [plus or minus] 1); Silver 107.8682 [plus or minus] 3 (107.868 [plus or minus] 1); Lutetium 174.967 [plus or minus] 1 (174.967 [plus or minus] 3). These values are incorporated in the Table of Standard Atomic Weights of the Elements 1981. Whereas in the past, the Table indicated uncertainties as either 1 or 3 in the last place, other single-digit uncertainties will in the future be quoted when there is convincing evidence that by their use, a more precise standard atomic weight can be tabulated. Important changes in annotations and the wording of footnotes to this Table and the Table of Isotopic Compositions are discussed. Changes in A-r(E) values and in their estimated uncertainties in the period since 1969 are analysed. The Report includes for the first time a Table of Atomic Weights abbreviated to five significant figures in the expectation that changes in the tabulated data will rarely be needed. Attention is drawn to the possibility of materials being commercially available containing elements with unusual atomic weights due to the enrichment or depletion of isotopes by free radical magnetic effects. The Relative Atomic Masses for Selected Radioisotopes are also tabulated.