  By means of $\epsilon$ and large $N$ expansions, we study generalizations of
the $O(N)$ model where the fundamental fields are tensors of rank $r$ rather
than vectors, and where the global symmetry (up to additional discrete
symmetries and quotients) is $O(N)^r$, focusing on the cases $r\leq 5$. Owing
to the distinct ways of performing index contractions, these theories contain
multiple quartic operators, which mix under the RG flow. At all large $N$ fixed
points, melonic operators are absent and the leading Feynman diagrams are
bubble diagrams, so that all perturbative fixed points can be readily matched
to full large $N$ solutions obtained from Hubbard-Stratonovich transformations.
The family of fixed points we uncover extend to arbitrary higher values of $r$,
and as their number grows superexponentially with $r$, these theories offer a
vast generalization of the critical $O(N)$ model.
  We also study sextic $O(N)^r$ theories, whose large $N$ limits are obscured
by the fact that the dominant Feynman diagrams are not restricted to melonic or
bubble diagrams. For these theories the large $N$ dynamics differ qualitatively
across different values of $r$, and we demonstrate that the RG flows possess a
numerous and diverse set of perturbative fixed points beginning at rank four.
