After reviewing the definition of stated skein modules for surfaces and 3-manifolds, I will detail how this recent notion allows us to relate topological constructions (related to cut and paste techniques) to algebraic ones (such as braided tensor products of algebra objects in braided categories). I will explain how the stated skein algebra of some special surfaces provides a topological description for some notable algebras (e.g. the quantised function ring $O_q(\mathfrak{sl}_2)$ or its "transmutation" $BSL_2(q)$). Then I will describe how stated skein moduli of 3-manifolds fit into a TQFT framework, albeit not a completely standard one. If time permits I will also discuss some unexpected noninjectivity results in dimension 3. This is joint work with Thang Le.