  While most strong-gravitational-lensing systems may be roughly modelled by a
single massive object between the source and the observer, in the details all
the structures near the light path contribute to the observed images. These
additional contributions, known as line-of-sight effects, are non-negligible in
practice. This article proposes a new theoretical framework to model the
line-of-sight effects, together with very promising applications at the
interface of weak and strong lensing. Our approach relies on the dominant-lens
approximation, where one deflector is treated as the main lens while the others
are treated as perturbations. The resulting framework is technically simpler to
handle than the multi-plane lensing formalism, while allowing one to
consistently model any sub-critical perturbation. In particular, it is not
limited to the usual external-convergence and external-shear parameterisation.
As a first application, we identify a specific notion of line-of-sight shear
that is not degenerate with the ellipticity of the main lens, and which could
thus be extracted from strong-lensing images. This result supports and improves
the recent proposal that Einstein rings might be powerful probes of cosmic
shear. As a second application, we investigate the distortions of
strong-lensing critical curves under line-of-sight effects, and more
particularly their correlations across the sky. We find that such correlations
may be used to probe, not only the large-scale structure of the Universe, but
also the dark-matter halo profiles of strong lenses. This last possibility
would be a key asset to improve the accuracy of the measurement of the
Hubble-Lema\^itre constant via time-delay cosmography.
