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.