We give sufficient conditions for the universality of tensor products $\{ T_n \comp{\otimes} R_n : n \in \N\}$ of sequences of operators defined on Fr\'{e}chet spaces. In particular we study when the tensor product $T\comp{\otimes}R$ of two operators is chaotic in the sense of Devaney. Applications are given for natural operators on function spaces of several variables, in Infinite Holomorphy, and for multiplication operators on the algebra $L(E)$ following the study of Kit Chan.