  We show that to each symmetric elliptic operator of the form \[ \mathcal{A} =
- \sum \partial_k \, a_{kl} \, \partial_l + c \] on a bounded Lipschitz domain
$\Omega \subset \mathbb{R}^d$ one can associate a self-adjoint
Dirichlet-to-Neumann operator on $L_2(\partial \Omega)$, which may be
multi-valued if 0 is in the Dirichlet spectrum of $\mathcal{A}$. To overcome
the lack of coerciveness in this case, we employ a new version of the
Lax--Milgram lemma based on an indirect ellipticity property that we call
hidden compactness. We then establish uniform resolvent convergence of a
sequence of Dirichlet-to-Neumann operators whenever their coefficients converge
uniformly and the second-order limit operator in $L_2(\Omega)$ has the unique
continuation property. We also consider semigroup convergence.
