  The main objective of this paper is to systematically develop a spectral and
scattering theory for selfadjoint Schr\"odinger operators with
$\delta$-interactions supported on closed curves in $\mathbb R^3$. We provide
bounds for the number of negative eigenvalues depending on the geometry of the
curve, prove an isoperimetric inequality for the principal eigenvalue, derive
Schatten--von Neumann properties for the resolvent difference with the free
Laplacian, and establish an explicit representation for the scattering matrix.
