  Boundedness of weighted composition operators $W_{u,\varphi}$ acting on the
classical Dirichlet space $\mathcal{D}$ as $W_{u,\varphi}f= u\, (f\circ
\varphi)$ is studied in terms of the multiplier space associated to the symbol
$\varphi$, i.e., ${\mathcal{M}(\phi)}=\{ u \in {\mathcal D}: W_{u,\phi} \hbox{
is bounded on } {\mathcal D} \}$. A prominent role is played by the multipliers
of the Dirichlet space. As a consequence, the spectrum of $W_{u,\varphi}$ in
$\mathcal{D}$ whenever $\varphi$ is an automorphism of the unit disc is
studied, extending a recent work of Hyv\"arinen, Lindstr\"om, Nieminen and
Saukko to the context of the Dirichlet space.
