I will discuss the Hopf algebraic (quantum symmetry) structures underlying certain four-dimensional finite quantum field theories which are related to the N=4 SYM theory by marginal deformations. The motivation is to understand why these theories, despite being finite (like N=4 SYM), are generically not integrable in the planar limit (unlike N=4 SYM). Apart from a better understanding of integrability, these hidden symmetries might eventually provide insight into the construction of the (still unknown) dual AdS/CFT geometries for these field theories.