  Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials
supported on a smooth compact hypersurface are defined explicitly via boundary
conditions. The spectral properties of these operators are investigated,
regularity results on the functions in their domains are obtained, and
analogues of the Birman--Schwinger principle and a variant of Krein's formula
are shown. Furthermore, Schatten--von Neumann type estimates for the
differences of the powers of the resolvents of the Schr\"odinger operators with
$\delta$ and $\delta'$-potentials, and the Schr\"odinger operator without a
singular interaction are proved. An immediate consequence of these estimates is
the existence and completeness of the wave operators of the corresponding
scattering systems, as well as the unitary equivalence of the absolutely
continuous parts of the singularly perturbed and unperturbed Schr\"odinger
operators. In the proofs of our main theorems we make use of abstract methods
from extension theory of symmetric operators, some algebraic considerations and
results on elliptic regularity.
