Tranforming a fixed random variable into an arbitrary one

Question

Say I have two random variables $Z$ and $X$ taking values in $\mathbb{R}^d$ and $\mathbb{R}^n$ both with continuous cdf. Is there a theorem that guarantees the existence of a continuous function $G:\mathbb{R^d}\rightarrow\mathbb{R}^n$ such that $G(Z)$ has the same distribution as $X$? I found a very involved explicit construnction but would be interested in an easy to apply theorem which leads to the same result.