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Based on a joint work with L. Martirosyan. We give a complete classification of ribbon tensor categories whose fusion ring is the one of the Lie algebra of type $G_2$, or of an associated fusion category of type $G_{2,k}$, for $k>-3$. Here $k$ is 3 times what is usually called the level. We get the expected result that they have to come from quantum groups. There is only a minor subtlety for $k=9$ (level 3), due to nontrivial automorphisms of the fusion ring. The main technical result is a classification of all simple representations of the braid group $B_4$ for which the image of a braid generator satisfies a cubic equation. This also solves a problem in the representation theory of cubic Hecke algebras.