AFP
The talks, held in Brussels, continued until the early hours of Friday morning without candidates being finalised.
On top of trying to find majorities for successors, leaders were also trying to decide what issues the EU should prioritise over the next five years.
A new summit will now be held on 30 June - just two days before a key meeting of the European Parliament.
All of the bloc's top jobs are changing hands this year, following European-wide elections in May.
Tense differences are said to have emerged over the successors for the key roles.
What roles need to be decided?
After Thursday's summit, European Council President Donald Tusk confirmed a new summit was necessary.
The most powerful role being decided is EU Commission president, held by ex-Luxembourg PM Jean-Claude Juncker since 2014.
The European Commission drafts EU laws, oversees national budgets, enforces EU treaties and negotiates international trade deals.
"I note with certain amusement that it is not easy to replace me," Mr Juncker told journalists at a press conference.
Besides the replacements for him and Mr Tusk, the leaders also have to decide on a new European Central Bank president and a high representative for foreign policy.
Nominees require the support of least 21 of the 28 EU leaders and a majority in the European Parliament - which will meet on 2 July.
Who is in for the top jobs?
The choices for the EU's top jobs have to take account of last month's European elections and achieve an acceptable balance regarding large and small states, gender and geography.
The elections produced more political fragmentation, including a surge in the numbers of Green and liberal MEPs.
The contest for a new Commission president began with the parliament's "Spitzenkandidat" (lead candidate) race, but leaders do not have to go along with the result.
The parliament says the president should come from the party with the most seats - which would mean Manfred Weber, a German in the centre-right European People's Party (EPP).
Strong potential rivals to Mr Weber are EU veteran Frans Timmermans (Dutch, from the centre-left Socialist bloc) and Danish EU commissioner Margrethe Vestager, a liberal.
But French President Emmanuel Macron is reported to favour Michel Barnier - the man who has been in charge of negotiating Brexit - for the post.
REUTERS
German Chancellor Angela Merkel had stressed the importance of finding a "joint solution" at the Council summit.
Leaving the meeting on Friday, Mr Macron said no consensus was found on any of the leading candidates.
"The three Spitzenkandidat, the three names were tested by Donald Tusk and he considered that they had found no majority on either of these three names," he said.
However, Estonian Prime Minister Juri Ratas struck a more upbeat note, telling journalists: "All the names are still on the table but I am positive we will find a solution next Sunday".
What other issues were on the table?
Climate change was a big issue in last month's European elections, but on Thursday the leaders failed to get unanimous agreement on a 2050 target date for going "carbon neutral".
France, Germany and the Netherlands were among a large group of states urging the EU to set that goal. It means reducing greenhouse gas emissions to a level where they are balanced by green initiatives, such as tree planting.
Poland, the Czech Republic and Hungary, heavily reliant on coal-fired power plants, opposed it. They are worried about the financial cost of shifting to green energy.
The date was therefore relegated to a footnote in the conclusions, which says: "For a large majority of member states, climate neutrality must be achieved by 2050."
Brexit is not a major issue at this summit, but on Friday Mr Tusk is to give a brief update on developments regarding the UK's departure.
The official EU position is that the Withdrawal Agreement negotiated with outgoing PM Theresa May - and rejected by the UK Parliament - will not be reopened.
When asked if Brexit negotiations could be extended again beyond the 31 October deadline, Irish Taoiseach (Prime Minister) Leo Varadkar was very sceptical.
"There's very much a strong view that there shouldn't be any more extensions. There's enormous hostility towards extension amongst the EU. I have endless patience but my colleagues do not," he said.
EU's Strategic Agenda
The summit adopted a Strategic Agenda for 2019-2024. It highlights:
- more intensive efforts to counter climate change
- measures to boost security on the EU's external borders and control migration into the bloc
- more resources to fight disinformation and "hybrid threats" which undermine European democracy
- better co-ordinated industrial policy, research and investment targeting digital technologies.
https://www.bbc.com/news/world-europe-48706193
