  The paper deals with existence and multiplicity of solutions of the
fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic
potential. As a consequence, the results can be applied to the special case
\begin{equation*} (a+b[u]_{s,A}^{2\theta-2})(-\Delta)_A^su+V(x)u=f(x,|u|)u\,\,
\quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$,
$a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $\theta\in[1,N/(N-2s))$,
$A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential,
$V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-\Delta
)_A^s$ is the fractional magnetic operator. In the super- and sub-linear cases,
the existence of least energy solutions for the above problem is obtained by
the mountain pass theorem, combined with the Nehari method, and by the direct
methods respectively. In the superlinear-sublinear case, the existence of
infinitely many solutions is investigated by the symmetric mountain pass
theorem.
