  We review recent developments and applications of the classification of the
Weyl tensor in higher dimensional Lorentzian geometries. First, we discuss the
general setup, i.e. main definitions and methods for the classification, some
refinements and the generalized Newman-Penrose and Geroch-Held-Penrose
formalisms. Next, we summarize general results, such as a partial extension of
the Goldberg-Sachs theorem, characterization of spacetimes with vanishing (or
constant) curvature invariants and the peeling behaviour in asymptotically flat
spacetimes. Finally, we discuss certain invariantly defined families of metrics
and their relation with the Weyl tensor classification, including: Kundt and
Robinson-Trautman spacetimes; the Kerr-Schild ansatz in a constant-curvature
background; purely electric and purely magnetic spacetimes; direct and (some)
warped products; and geometries with certain symmetries. To conclude, some
applications to quadratic gravity are also overviewed.
