  We study a connection between random tensors and random matrices through
$U(\tau)$ matrix models which generate fully packed, oriented loops on random
surfaces. The latter are found to be in bijection with a set of regular
edge-colored graphs typically found in tensor models. It is shown that the
expansion in the number of loops is organized like the 1/N expansion of
rank-three tensor models. Recent results on tensor models are reviewed and
applied in this context. For example, configurations which maximize the number
of loops are precisely the melonic graphs of tensor models and a scaling limit
which projects onto the melonic sector is found. We also reinterpret the double
scaling limit of tensor models from the point of view of loops on random
surfaces. This approach is eventually generalized to higher-rank tensor models,
which generate loops with fugacity $\tau$ on triangulations in dimension $d-1$.
