  We study the spectrum of the Robin Laplacian with a complex Robin parameter
$\alpha$ on a bounded Lipschitz domain $\Omega$. We start by establishing a
number of properties of the corresponding operator, such as generation
properties, local analytic dependence of the eigenvalues and eigenspaces on
$\alpha \in \mathbb C$, and basis properties of the eigenfunctions. Our focus,
however, is on bounds and asymptotics for the eigenvalues as functions of
$\alpha$: we start by providing estimates on the numerical range of the
associated operator, which lead to new eigenvalue bounds even in the case
$\alpha \in \mathbb R$. For the asymptotics of the eigenvalues as $\alpha \to
\infty$ in $\mathbb C$, in place of the min-max characterisation of the
eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the
real case, we exploit the duality between the eigenvalues of the Robin
Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to
show that every Robin eigenvalue either diverges to $\infty$ in $\mathbb C$ or
converges to a point in the spectrum of the Dirichlet Laplacian, and also to
give a comprehensive treatment of the special cases where $\Omega$ is an
interval, a hyperrectangle or a ball. This leads to the conjecture that on a
general smooth domain in dimension $d\geq 2$ all eigenvalues converge to the
Dirichlet spectrum if ${\rm Re}\, \alpha$ remains bounded from below as $\alpha
\to \infty$, while if ${\rm Re}\, \alpha \to -\infty$, then there is a family
of divergent eigenvalue curves, each of which behaves asymptotically like
$-\alpha^2$.
