A highly branched polymer comprising repeating units which each have an acid group such as sulfo group, said repeating units being represented by formula [1] or the like, and a dispersant for carbon nanotubes (CNTs) which comprises the highly branched polymer can disperse CNTs in a medium such as an organic solvent to the individual sizes and can yield thin films having improved conductivity. In formula [1], any one of A1 to A5 is a sulfo group, and the others are each a hydrogen atom.