  Given a complex, elliptic coefficient function we investigate for which
values of $p$ the corresponding second-order divergence form operator,
complemented with Dirichlet, Neumann or mixed boundary conditions, generates a
strongly continuous semigroup on $\mathrm{L}^p(\Omega)$. Additional properties
like analyticity of the semigroup, $\mathrm{H}^\infty$-calculus and maximal
regularity are also discussed. Finally we prove a perturbation result for real
coefficients that gives the whole range of $p$'s for small imaginary parts of
the coefficients. Our results are based on the recent notion of
$p$-ellipticity, reverse H\"older inequalities and Gaussian estimates for the
real coefficients.
