  The spectral properties of non-self-adjoint extensions $A_{[B]}$ of a
symmetric operator in a Hilbert space are studied with the help of ordinary and
quasi boundary triples and the corresponding Weyl functions. These extensions
are given in terms of abstract boundary conditions involving an (in general
non-symmetric) boundary operator $B$. In the abstract part of this paper,
sufficient conditions for sectoriality and m-sectoriality as well as sufficient
conditions for $A_{[B]}$ to have a non-empty resolvent set are provided in
terms of the parameter $B$ and the Weyl function. Special attention is paid to
Weyl functions that decay along the negative real line or inside some sector in
the complex plane, and spectral enclosures for $A_{[B]}$ are proved in this
situation. The abstract results are applied to elliptic differential operators
with local and non-local Robin boundary conditions on unbounded domains, to
Schr\"odinger operators with $\delta$-potentials of complex strengths supported
on unbounded hypersurfaces or infinitely many points on the real line, and to
quantum graphs with non-self-adjoint vertex couplings.
