  By employing a suitable multiplicative It\^o noise with radial structure and
with more than linear growth, we show the existence of a unique,
global-in-time, strong solution for the stochastic Euler equations in two and
three dimensions. More generally, we consider a class of stochastic partial
differential equations (SPDEs) with a superlinear growth drift and suitable
nonlinear, multiplicative It\^o noise, with the stochastic Euler equations as a
special case within this class. We prove that the addition of such a noise
effectively prevents blow-ups in the solution of these SPDEs.
