On the British side, Davis will take the lead, reporting to May. Britain's ambassador to the EU, Tim Barrow, will also play a major role, and the Foreign Office will talk to individual member states to try to get them on its side.

On the EU side, it's complicated. As Britain's Institute for Government recently pointed out, "the U.K. is negotiating with 27 member states, not a unified bloc."

French diplomat Michel Barnier is the chief negotiator for the European Commission, the bloc's executive arm. He'll receive direction from the Council, which represents the leaders of the member states.

The European Parliament also wants a say, and will have to approve the final deal between Britain and the bloc.