It’s inauguration day in the US and European leaders will be among the throngs of western watchers welcoming the sight of Joe Biden being sworn into office.

The Trump years have provoked deep soul-searching in Brussels and other EU capitals about the transatlantic relationship and the union’s place in the world. It has lodged the language of “strategic autonomy” and “European sovereignty” firmly in the EU’s vernacular — rhetoric that will not be disappearing under the new presidency.

A new poll from the European Council on Foreign Relations, which surveyed 15,000 Europeans in 11 countries, shows that despite relief at Mr Trump’s exit, it won’t be business as usual for the EU with Mr Biden. Nearly a third of Europeans think US voters still “can’t be trusted” after the Trump years, with the greatest level of mistrust among Germans. It also shows that on the whole, Europeans are more positive about themselves and the EU’s political system than in previous years.

The survey makes for interesting reading for European leaders about the state of public opinion. Here are the main themes set to define EU-US relations during the early months of the Biden administration.

Climate The new president will use his inauguration to declare the US a signatory to the Paris climate change accord — reversing Mr Trump’s executive order to withdraw from the agreement in 2017. US climate diplomacy under Mr Biden will represent a major break from the previous administration, and one that will be heartily welcomed by the EU, which has spent the past year forging its role as an international leader on curbing emissions. Mr Biden wants to use his first year in office to pass a net zero emissions target by 2050 — the EU’s own aim. He has also discussed implementing a carbon levy at US frontiers, an idea with parallels in Europe’s carbon border adjustment to be proposed by Brussels later this year.

The appointment of John Kerry as Washington’s climate envoy has also been roundly welcomed in Brussels. Mr Kerry will hold one of his first calls with Frans Timmermans, the EU’s vice-president for the Green Deal, on Thursday.

Tech and antitrust Mr Trump trashed Margrethe Vestager, the commission’s vice-president for competition and digital policy, as hating the US “perhaps worse than any person I’ve ever met”. The Danish commissioner had never met Mr Trump, but that is beside the point as the incoming administration could provide Brussels with much needed co-operation to go after Big Tech.

Speaking to the FT this week, Ms Vestager said: “We have a lot of things on our to-do list that we would want to discuss with the incoming presidency in the new administration. We are hoping for a constructive debate so that we can stay aligned.”

On digital tax, the Trump administration has accused EU countries, particularly France, of unfairly targeting American companies, but held off on applying tariffs to French goods in a bid to quell transatlantic trade tensions last year. Still, it remains unclear what position the new president will take on digital tax. Some clarity is expected within the coming months.

Trade After a stormy trade relationship with Mr Trump, Brussels is now counting on a Biden-induced thaw. Mr Trump pursued a protectionist agenda aimed at prioritising US workers that involved, among other things, the imposition of tariffs on European steel and aluminium in 2018. He also declined to make common cause with the EU on relations with China, preferring to pursue his own unilateral measures.

Mr Biden will need to tread carefully as he recasts trade relations given the need to reassure US manufacturing workers in the Midwest and elsewhere that he will defend their interests.

Among the most immediate questions is the two sides’ long-running dispute over civil aviation. The EU last year slapped duties on $4bn of US imports including aircraft in retaliation for injury caused by illegal subsidies to Boeing. The move came after the US imposed duties on $7.5bn of imports from the EU, including Airbus aircraft and Scotch whisky.

Valdis Dombrovskis, the EU trade commissioner, said on Tuesday that Brussels will be “working closely with the Biden administration” on the file, adding that both sides need to either withdraw or suspend tariffs. He expressed optimism about the arrival of the new US president, saying Mr Biden is a “great supporter of multilateralism and of international alliances”.

Foreign policy Mr Biden has more room to manoeuvre on foreign policy than in some other areas — but the transatlantic to-do list is formidable. A top priority will be to revive the Iran nuclear deal, which the president-elect has signalled he wants to rejoin — provided Tehran honours its commitments. Russia will also loom large, especially after the Sunday’s detention of opposition activist Alexei Navalny.

But the biggest focus and potential source of transatlantic tension will be China, especially after the EU irritated the incoming administration in Washington by signing an investment agreement with Beijing a few weeks ago. On defence, Mr Biden won’t be as publicly rude as Mr Trump was about European countries’ military spending — but he’s equally unlikely to ease longstanding US pressure for Nato allies to spend more.

European domination of UK trade has been the norm in peacetime. Chart showing UK trade with Empire/Commonwealth and with US, as a % of trade with current EU countries

Europe will continue to dominate UK trade for decades as it has throughout the postwar era, as shown in this chart from the FT. It features in Philip Stephens’ exploration of Britain’s search for a place in the world — economically and diplomatically — for the past 70 years.

In Brussels, the formal end of Brexit talks has led former chief negotiator Michel Barnier to be appointed as a special adviser to commission chief Ursula von der Leyen. It is the third time the Frenchman has served in such a role.

Commission president Ursula von der Leyen and Portugal’s prime minister António Costa take part in a European parliament debate to mark the start of the Portuguese EU presidency (10.15 CET).