Matrix Bases for Star Products: a Review

We review the matrix bases for a family of noncommutative $\star$ products based on a Weyl map. These products include the Moyal product, as well as the Wick–Voros products and other translation invariant ones. We also review the derivation of Lie algebra type star products, with adapted matrix bases. We discuss the uses of these matrix bases for field theory, fuzzy spaces and emergent gravity.