The problem of finding complete metrics with constant scalar curvature is fundamental in Riemannian and conformal geometry. In this talk, motivated by the analogy between conformal and CR geometry, we study the existence of complete contact forms with constant Tanaka–Webster scalar curvature. We prove that, under mild assumptions, the universal cover of a compact strictly pseudoconvex CR manifold whose fundamental group has infinite profinite completion admits infinitely many pairwise non-homothetic contact forms of this type. As applications, we consider complements of real or complex spheres in the standard CR sphere, as well as circle bundles over compact Kähler manifolds. This is based on joint work with Jeffrey S. Case.