  Ordinary tensor models of rank $D\geq 3$ are dominated at large $N$ by
tree-like graphs, known as melonic triangulations. We here show that
non-melonic contributions can be enhanced consistently, leading to different
types of large $N$ limits. We first study the most generic quartic model at
$D=4$, with maximally enhanced non-melonic interactions. The existence of the
$1/N$ expansion is proved and we further characterize the dominant
triangulations. This combinatorial analysis is then used to define a
non-quartic, non-melonic class of models for which the large $N$ free energy
and the relevant expectations can be calculated explicitly. They are matched
with random matrix models which contain multi-trace invariants in their
potentials: they possess a branched polymer phase and a 2D quantum gravity
phase, and a transition between them whose entropy exponent is positive.
Finally, a non-perturbative analysis of the generic quartic model is performed,
which proves analyticity in the coupling constants in cardioid domains.
