"Symmetric algebra" is a fancy way of saying "polynomial ring." The symmetric algebra of a k-vector space V is the enveloping commutative algebra of V in the category kVect, and can be realized as the polynomial ring generated by any basis of V. Quantum symmetric algebras are analogues of polynomial rings in the category of modules over the quantized universal enveloping algebra of a semisimple Lie algebra. Familiar examples include quantum polynomial and matrix algebras as well as coordinate algebras of quantum Euclidean and symplectic vector spaces, but there are more exotic ones also. I will describe the general construction of quantum symmetric algebras and show that they satisfy a universal mapping property analogous to the one for ordinary symmetric algebras. This requires an appropriate notion of commutativity for algebras in Uq(g)-Mod. I will try to illustrate the general theory with simple examples.