Modelling questions for transitive quantum groups

This is a survey on the transitive quantum groups $G\subset S_N^{+}$, and on the flat matrix models $\pi :C(G)\to M_N(C(X))$ for the corresponding Hopf algebras. We review the known results on the subject, with a number of improvements, technical generalizations, and some new results as well, and we discuss the open problems.