  We characterize those pairs $(\psi,\varphi)$ of smooth mappings
$\psi:\mathbb{R}^d\rightarrow\mathbb{C},\varphi:\mathbb{R}^d\rightarrow\mathbb{R}^d$
for which the corresponding weighted composition operator
$C_{\psi,\varphi}f=\psi\cdot(f\circ\varphi)$ acts continuously on
$\mathscr{S}(\mathbb{R}^d)$. Additionally, we give several easy-to-check
necessary and sufficient conditions of this property for interesting special
cases. Moreover, we characterize power boundedness and topologizablity of
$C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R}^d)$ in terms of $\psi,\varphi$.
Among other things, as an application of our results we show that for a
univariate polynomial $\varphi$ with $\text{deg}(\varphi)\geq 2$, power
boundedness of $C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R})$ for every
$\psi\in\mathscr{O}_M(\mathbb{R})$ only depends on $\varphi$ and that in this
case power boundedness of $C_{\psi,\varphi}$ is equivalent to
$(C_{\psi,\varphi}^n)_{n\in\mathbb{N}}$ converging to $0$ in
$\mathcal{L}_b(\mathscr{S}(\mathbb{R}))$ as well as to the uniform mean
ergodicity of $C_{\psi,\varphi}$. Additionally, we give an example of a power
bounded and uniformly mean ergodic weighted composition operator
$C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R})$ for which neither the
multiplication operator $f\mapsto \psi f$ nor the composition operator
$f\mapsto f\circ\varphi$ acts on $\mathscr{S}(\mathbb{R})$. Our results
complement and considerably extend various results of Fern\'andez, Galbis, and
the second named author.
