  The melonic limit is a relatively new type of large-$N$ limit, differing from
the much older and well-known large-$N$ limits of vector and matrix field
theories, which are dominated by cactus and planar Feynman diagrams,
respectively. The melonic limit typically appears in tensor field theories,
characterized by an invariance group in which the fields transform as the
product of $r\geq 3$ fundamental representations of $r$ different simple Lie
groups. As the name suggests, in such a limit the perturbative expansion of
free energy and correlators are dominated by melonic diagrams. The latter form
a manageable subset of the planar diagrams, but with a richer structure than
cactus diagrams, and therefore they open the possibility of studying in a
controlled manner new types of fixed points of the renormalization group. We
call \emph{melonic conformal field theories (CFTs)} those fixed-point theories
that are found in the melonic limit. We concisely review the construction and
analysis of tensor field theories in $d\geq 2$ (Euclidean) spacetime
dimensions, with special emphasis on the general theoretical framework, and on
specific results for the fixed points of some models.
