Considerations of the non-abelian Radon transform were started in [Manakov, Zakharov, 1981] in the framework of the theory of solitons in dimension 2+1. On the other hand, the problem of inversion of transforms of such a type arises in different tomographies, including emission tomographies, polarization tomographies, and vector field tomography. In this article we give a short review of old and recent results on this subject. This article is an extended version of the talk given at the conference "100 Years of the Radon Transform", Linz, 27-31 March 2017.