There is no unique way to generalize the mass operator 0 2 −p · p to curved spacetimes. The possible generalizations using either an analytic or an algebraic (group-theoretic) approach are discussed. We investigate in detail three possibilities: (i) the generalized D'Alembertian $$- \tilde \square = - (1/\sqrt { - g} )(\partial /\partial x^v )(\sqrt { - g} g^{\mu v} {\text{ }}\partial /\partial x^\mu$$ (ii) the operator ( $$- (\tilde \square + \tfrac{1}{6}{\text{R)}}$$ , which is conformally invariant in the massless case, and (iii) an operator suggested by Meggs for Robertson-Walker spacetimes. A criterion for choosing among the different possibilities is proposed. It is used to eliminate the Meggs operator, but is not stringent enough to decide between the other two.