  This paper investigates the stability properties of the spectrum of the
classical Steklov problem under domain perturbation. We find conditions which
guarantee the spectral stability and we show their optimality. We emphasize the
fact that our spectral stability results also involve convergence of
eigenfunctions in a suitable sense according with the definition of connecting
system by \cite{Vainikko}. The convergence of eigenfunctions can be expressed
in terms of the $H^1$ strong convergence. The arguments used in our proofs are
based on an appropriate definition of compact convergence of the resolvent
operators associated with the Steklov problems on varying domains. In order to
show the optimality of our conditions we present alternative assumptions which
give rise to a degeneration of the spectrum or to a discontinuity of the
spectrum in the sense that the eigenvalues converge to the eigenvalues of a
limit problem which does not coincide with the Steklov problem on the limiting
domain.
