(joint with Ehud Hrushovski) We will give a structure theory for groups interpretable in ACVF, decomposing them in term of groups internal to the residue field, groups internal to the value group and group schemes over the valuation ring. Groups with a stably dominated type stable by translation (stably dominated groups) play an important role in this structure theory: our main result is that Abelian groups have a maximal value group internal quotient whose kernel is covered by stably dominated groups. We also relate stably dominated groups to group schemes over the valuation ring. Finally, we will use this structure theory to show that, up to definable isomorphism, every field definable in ACVF is either the residue field or the valued field itself.