The existence of the dielectric constant epsilon is investigated for fluids composed of classical deformable (polarizable) molecules. The development is based upon generalized functional-derivative relations which involve joint distributions in molecular positions r/sub k/ and dipole moments ..mu../sub k/. Sufficient conditions for the existence of epsilon are expressed in terms of the generalized direct correlation function c(12) = c(r/sub 1/, ..mu../sub 1/; r/sub 2/, ..mu../sub 2/). It is found that epsilon exists if -kTc(12) depends only on relative positions and dipole moment directions (in addition to Vertical Bar..mu../sub 1/Vertical Bar and Vertical Bar..mu../sub 2/Vertical Bar), and becomes asymptotic to the dipole--dipole potential at long range. An expression for epsilon in terms of a short-ranged total correlation function h/sub 0/(12) emerges automatically from the development. An expression for epsilon in terms of c(12) is also derived. The latter expression involves an inverse kernel in (Vertical Bar..mu../sub 1/Vertical Bar, Vertical Bar..mu../sub 2/Vertical Bar) space. The case of rigid polar molecules is reconsidered as a special case of the present formulation.