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.

29 pages, 10 figures. Contribution to the proceedings of the Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2019), 31 August - 25 September 2019, Corfu, Greece