Part I. Seven of the double sulphates and selenates ('Tutton salts') of copper have been examined by the method of paramagnetic resonance. The deviation of the effective gyromagnetic ratio from the free-spin value of 2 is shown to be generally consistent with the theory of Polder (1942), who assumes a crystalline electric field of tetragonal symmetry to act on each ion. In some salts an appreciable departure from tetragonal symmetry is observed. Part II. The variation of the width of the absorption lines with the direction of the applied magnetic field is studied and compared with that calculated from magnetic dipole interaction. The line shapes show that exchange interaction is present in varying degree, though much less powerful than in copper sulphate. The 'mean square moment' of the lines can be explained by the magnetic dipole interaction, together with a contribution from a hyperfine structure which is not resolved except in a highly diluted salt.