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Naturally the UK, or Westminster at least, is wrapped up in the race for who will become the next prime minister.

And, naturally, EU leaders are looking on with trepidation at who might succeed Mrs May.

But there's another leadership race they are much more focused on. Several in fact.

At the end of October, Jean-Claude Juncker's term as President of the European Commission will come to an end.

His 28 commissioners will retire from their posts and probably head back to where they all came from - national politics or diplomacy.

The European Central Bank (ECB) will get a new chief, as will the European External Action Service (EEAS - the EU's Foreign Office) as Mario Draghi and Federica Mogherini's terms both come to an end.

Donald Tusk will relinquish the post of European Council President and the Parliament too will replace President Antonio Tajani.

It is all change in the European Union machine and the mechanics is complicated.

It involves a good dose of horse trading, a democratic legitimacy that some will inevitably question and a few artful manoeuvres. And it all begins in earnest at tonight's European Council summit.

The key word is spitzenkandidat - the name for a system introduced in 2014 in an attempt to make the EU more democratic and accountable.

Prior to that, the president of the EU Commission (which proposes EU legislation) was nominated by European Council (made up of the elected leaders of the EU member countries). Under the spitzenkandidat system, the political groups of the European Parliament put forward candidates and the lead candidate (spitzenkandidat) gets the job.

However, to secure that position, the successful candidate needs the backing of a qualified majority of the European Council (the elected EU leaders) and a majority from the European Parliament (the 751 elected MEPs).

The official lead candidate is German MEP Manfred Weber because the political group to which his German party is associated (the centre right EPP) is the largest in the parliament.

But, despite being backed by German Chancellor Merkel, Mr Weber isn't all that popular among the EU leaders who also don't hugely like the spitzenkandidat process either.

Here's where the horse trading comes in. Different EU countries have different visions for the EU. If they can get their candidate into the top job in the Commission then their vision can be realised.

It is perfectly possible that none of the candidates put forward from each of the European Parliament political groups actually gets the job.

French President Emanuel Macron, who is pushing for liberal-leaning reforms and further EU integration, doesn't like Mr Weber and his centre-right vision.

Mr Weber may also struggle to get a majority in the Parliament because the weekend's European Parliament elections gave a boost for the liberal ALDE group and the Greens too.

So, at tonight's summit, the outgoing council president Mr Tusk will try to begin to find some consensus among the leaders.

Is there a candidate they can agree on? Is there a compromise candidate? If certain countries are satisfied with their person in another job - high representative in the EEAS or head of the ECB then can consensus be found that way?

Tory leadership race: The runners and riders vying to replace Theresa May

Sky News takes a look at the Conservatives who are trying to get their way in to Number 10.

The names to watch out for over the next few weeks:

:: For Commission President

• Mr Weber as mentioned above
• Michel Barnier (the EU Brexit Negotiator and a Macron favourite)
• Margrethe Vestager (outgoing Competition Commissioner & ALDE/liberal spitzenkandidat)
• Franz Timmermans (outgoing vice-President of Commission and S&D/centre-left spitzenkandidat
• Christine Lagarde (head of the International Monetary Fund)

:: For the Council President

• Dalia Grybauskaite (current President of Lithuania)
• Mark Rutte (current Prime Minister of Netherlands)
• Angela Merkel (current German Chancellor)

Why should all this be of particular interest to readers in the UK? Because whether Britain leaves the EU with a deal, without a deal or stay in, it will be the new guard in Brussels with whom the British government will be forced to deal.