We give a definition of mass for conformally compactifiable initial data sets. The asymptotic conditions are compatible with existence of gravitational radiation, and the compactifications are allowed to be polyhomogeneous. We show that the resulting mass is a geometric invariant, and we prove positivity thereof in the case of a spherical conformal infinity. When R(g) -- or, equivalently, trgK -- tends to a negative constant to order one at infinity, the definition is expressed purely in terms of three-dimensional or two-dimensional objects.