We prove the existence of global solutions to singular SPDEs on Rd with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions d=4, 5 and in the parabolic setting for d = 2, 3. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the phi d4 Euclidean quantum field theory via Parisi-Wu stochastic quantization, while the elliptic equations are linked to the phi d-24 Euclidean quantum field theory via the Parisi-Sourlas dimensional reduction mechanism.