  We review two known in the literature exemples of non-associative star
products. The first one is the phase space star product representing
quantization of non-geometric $R$-flux background in closed string theory. The
second is the octonionic star product which provides the quantization of the
quasi-Poisson algebra isomorphic to the Malcev algebra of imaginary octonions.
We discuss in details the construction, properties and physical applications of
these star products. In particular, we consider the quantization of
non-geometric M-theory background. Based on these two exemples we formulate the
minimal set of physically motivated conditions for the definition of
non-associative deformation quantization.
