The Department for International Trade has prioritised deals with the EU, the United States, Japan, Australia and New Zealand. Here's where things stand:
Total trade value: 232.7 billion pounds (1)
Both sides have said they want a quick deal, but there are clear hurdles. The United States elects a new president in November, and the UK says there is no set timetable because it does not want time pressure to be used against it.
Two rounds of talks have taken place so far, with the third expected to begin on Monday. Both sides have described the talks as constructive and positive, but there are underlying tensions.
The United States imposed retaliatory tariffs on EU exports, including some from Britain, after a row over aircraft subsidies. Britain has demanded that those tariffs, which apply to a wide range of goods including Scotch whisky and tailored suits, be removed to show goodwill.
Other high-profile issues include whether a deal will give U.S. firms access to Britain's state-funded healthcare system, and whether Britain will accept U.S. food hygiene standards. The British government says the National Health Service is not on the table, and food standards will not be lowered.
Total trade value: 673 billion pounds (1)
Britain and the European Union have held six formal rounds of talks on a trade agreement to form part of a deal on their future relationship when a status quo transition period expires at the end of this year.
Both sides agreed last month to speed up their talks but their positions are far apart, particularly on the "level playing field" guarantees of fair competition and on fisheries.
Britain said after the last round that an agreement could be reached in September; EU officials say the earliest any breakthrough could come is towards the end of August.
Total trade value: 18.6 billion pounds (1)
Britain said in its negotiating mandate, published on June 17, that it wanted an ambitious and comprehensive deal including a focus on technology, innovation and research and development.
The first round of talks was held remotely between June 29 and July 10. Trade minister Liz Truss said: "Discussions between negotiators were productive and reflected our shared ambition to secure a comprehensive deal to boost trade and investment between our like-minded economies."
The next round of talks is planned for September.
Total trade value: 31.6 billion pounds (1)
Negotiations began in June. Both sides have said they want a deal to come into force by the end of 2020.
Tokyo wanted to conclude talks by the end of July, but Truss has since said she will not be bounced into signing a deal because of time pressure.
Britain says any deal with Japan should be at least as good as the one signed with the EU, and that that deal can be used as a template to speed the process up.
British analysis estimated that, in the long run, the trade deal could increase annual trade flows between both countries by 15.2 billion pounds and increase the size of the UK economy by 1.5 billion pounds.
Total trade value: 2.9 billion pounds (1)
Britain said in its negotiating mandate, published on June 17, that it wanted an ambitious and comprehensive deal supporting both governments' 'Net Zero' commitments on climate change.
The first round of trade talks began via videoconference on July 13 and concluded on July 24. Further rounds are expected in due course, Britain said.
(1) Office for National Statistics, UK total trade; all countries, non-seasonally adjusted, 2019, as of July 23.
(Reporting by William James and Elizabeth Piper; Editing by Kevin Liffey)