  The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and
others) asks whether any metric of curvature $K\geq 0$ on the sphere is induced
on the boundary of a unique convex body in $\R^3$. The answer was extended to
surfaces in hyperbolic space by Alexandrov in the 1950s, and a ``dual''
statement, describing convex bodies in terms of the third fundamental form of
their boundary (e.g. their dihedral angles, for an ideal polyhedron) was later
proved.
  We describe three conjectural generalizations of the Weyl problem in $\HH^3$
and its dual to unbounded convex subsets and convex surfaces, in ways that are
relevant to contemporary geometry since a number of recent results and
well-known open problems can be considered as special cases. One focus is on
convex domain having a ``thin'' asymptotic boundary, for instance a quasicircle
-- this part of the problem is strongly related to the theory of Kleinian
groups. A second direction is towards convex subsets with a ``thick'' ideal
boundary, for instance a disjoint union of disks -- here one find connections
to problems in complex analysis, such as the Koebe circle domain conjecture. A
third direction is towards complete, convex disks of infinite area in $\HH^3$
and surfaces in hyperbolic ends -- with connections to questions on circle
packings or grafting on the hyperbolic disk. Similar statements are proposed in
anti-de Sitter geometry, a Lorentzian cousin of hyperbolic geometry where
interesting new phenomena can occur, and in Minkowski and Half-pipe geometry.
We also collect some partial new results mostly based on recent works.
