  We prove existence of global solutions to singular SPDEs on $\mathbb{R}^d$
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 $\Phi^4_d$ Euclidean quantum field theory via Parisi--Wu
stochastic quantization, while the elliptic equations are linked to the
$\Phi^4_{d-2}$ Euclidean quantum field theory via the Parisi--Sourlas
dimensional reduction mechanism.
