  Using covariant methods, we construct and explore the Wetterich equation for
a non-minimal coupling $F(\phi)R$ of a quantized scalar field to the Ricci
scalar of a prescribed curved space. This includes the often considered
non-minimal coupling $\xi \phi^2 R$ as a special case. We consider the
truncations without and with scale- and field-dependent wave function
renormalization in dimensions between four and two. Thereby the main emphasis
is on analytic and numerical solutions of the fixed point equations and the
behavior in the vicinity of the corresponding fixed points. We determine the
non-minimal coupling in the symmetric and spontaneously broken phases with
vanishing and non-vanishing average fields, respectively. Using functional
perturbative renormalization group methods, we discuss the leading universal
contributions to the RG flow below the upper critical dimension $d=4$.
