  This thesis focuses on renormalization of tensor field theories. Its first
part considers a quartic tensor model with $O(N)^3$ symmetry and long-range
propagator. The existence of a non-perturbative fixed point in any $d$ at large
$N$ is established. We found four lines of fixed points parametrized by the
so-called tetrahedral coupling. One of them is infrared attractive, strongly
interacting and gives rise to a new kind of CFT, called melonic CFTs which are
then studied in more details. We first compute dimensions of bilinears and OPE
coefficients at the fixed point which are consistent with a unitary CFT at
large $N$. We then compute $1/N$ corrections. At next-to-leading order, the
line of fixed points collapses to one fixed point. However, the corrections are
complex and unitarity is broken at NLO. Finally, we show that this model
respects the $F$-theorem. The next part of the thesis investigates sextic
tensor field theories in rank $3$ and $5$. In rank $3$, we found two IR stable
real fixed points in short range and a line of IR stable real fixed points in
long range. Surprisingly, the only fixed point in rank $5$ is the Gaussian one.
For the rank $3$ model, in the short-range case, we still find two IR stable
fixed points at NLO. However, in the long-range case, the corrections to the
fixed points are non-perturbative and hence unreliable: we found no precursor
of the large $N$ fixed point. The last part of the thesis investigates the
class of model exhibiting a melonic large $N$ limit. We prove that models with
tensors in an irreducible representation of $O(N)$ or $Sp(N)$ in rank $5$
indeed admit a large $N$ limit. This generalization relies on recursive bounds
derived from a detailed combinatorial analysis of Feynman graphs involved in
the perturbative expansion of our model.
