The Kaufman bracket skein algebra of a surface is one of few
constructions for which the relationship between quantum topology and
hyperbolic geometry is relatively concrete and well-understood.
Broadly speaking, the Kauffman bracket skein algebra is the quantum
analog of the space of hyperbolic metrics on a 3-dimensional
thickening of the surface. In this talk, we will survey two recent
generalizations, defined by Roger and Yang and by Muller, which
include both arcs and skeins on the surface and which can be
interpreted in terms of the decorated Teichmüller space.