“Thank you” – they were the two simple words at the heart of the message from world leaders to those who took part in the Normandy invasion that marked the beginning of the end of the Second World War.

Prime Minister Theresa May used them directly to the hundreds of veterans in Normandy as she marked the 75th anniversary of the D-Day landings.

And the sentiment was echoed by French President Emmanuel Macron, who said we owed them “our freedom”.

The day of commemorative events began with a lone Scots piper marking the moment the first UK soldiers went ashore. 

It also included special services of remembrance at Bayeux Cathedral and the nearby Commonwealth War Graves cemetery in northern France.

Donald Trump later told US veterans they were “the pride of the nation” at a service at a US war cemetery.

Mrs May and Mr Macron paid their respects at Ver-Sur-Mer, during the inauguration of the British Normandy Memorial. It overlooks Gold Beach where many of the troops arrived for what was the largest combined land, air and naval operation in history.

Funded by the Normandy Memorial Trust, the monument lists the names of all 22,442 members of the British armed forces who died in the Normandy campaign that began on June 6, 1944.

Mrs May told the ceremony that she was humbled to be able to mark the moment with veterans, who belonged to a “very special generation”.

“A generation whose unconquerable spirit shaped the post-war world. They didn’t boast. They didn’t fuss. They served,” she said.

“And they laid down their lives so that we might have a better life and build a better world.

“If one day can be said to have determined the fate of generations to come in France, in Britain, in Europe and the world, that day was June 6, 1944,” she added.

Mrs May, completing one of her last engagements as prime minister, added: “Standing here, as the waves wash quietly on to the shore, it’s almost impossible to grasp the raw courage that it must have taken that day to leap out from landing craft and into the surf – despite the fury of battle.”

She went on: “To our veterans here in Normandy today, I want to say the only words we can: thank you.”

Mr Macron also thanked “all those who were killed so that France could become free again”.

He said: “This is where young men, many of whom had never set foot on French soil, landed at dawn under German fire, risking their lives while fighting their way up the beach, which was littered with obstacles and mines.”

And, in a reference to ongoing uncertainty over Brexit, the French president went on to say he was proud to have worked with Mrs May.

“Leaders may come and go but their achievements remain. The force of our friendship will outlast current events,” he said.

Mr Macron was later joined by Donald Trump at a cemetery in Colleville-sur-Mer, in front of the beach codenamed Omaha for the landings.

Mr Macron took the opportunity to appeal directly to Donald Trump to fulfil the “promise of Normandy” by embracing pillars of the postwar peace such as the European Union and Nato.

Mr Macron told of his gratitude for the valour of US troops in the Second World War but also called for the White House to re-engage with the principles of multi-lateralism.

Speaking in front of 15,000 people gathered at the resting place for 9,387 US troops killed in the Normandy campaign, Mr Macron said: “Dear Donald Trump, the United States is never greater than when it is fighting for the freedom of others.

“The United States of America is never greater than when it shows its loyalty. 

“Loyalty to the universal values that the founding fathers defended, when nearly two and a half centuries ago France came to support its independence. What we owe you is to show ourselves worthy of the heritage of peace that you have left us.

“Being worthy of the promise of Normandy means never forgetting that free people, when they join forces, can surmount any adversity,” Macron went on.

“We shall never cease to perpetuate the alliance of free peoples. 

“That is what the victorious sides did, when they created the United Nations, the North Atlantic Treaty Organisation. That is what a few years later the leaders of Europe did in bringing about the European Union. The lessons of Colleville-sur-Mer are clear: liberty and democracy are inseparable.”

The US president praised America’s “exceptional might” and said it would be “forever be strong”. He said US soldiers who liberated France were “sustained by the confidence that America can do anything. The exceptional might came from a truly exceptional spirit”.

He paid tribute to a group of around 60 US veterans sitting in the front row and said those who battled on the beach “saved freedom”.

“You are among the very greatest Americans who will ever live. You are the pride of our nation. You are the glory of our republic and we thank you from the bottom of our hearts,” he said.

“Those who fought here won a future, they won the survival of our civilisation and they showed us the way to love, cherish and defend our way of life for many centuries to come.”

Earlier a service of remembrance was also held at the cathedral in Bayeux, the first city to be liberated by the invasion.

The 1,000-strong congregation, which included the Prince of Wales and Duchess of Cornwall,  Mrs May, Labour leader Jeremy Corbyn, Defence Secretary Penny Mordaunt, First Minister of Scotland Nicola Sturgeon and shadow foreign secretary Emily Thornberry, declared in unison: “We shall remember them.” A two-minute silence followed.

In the UK, the Duke of Cambridge delivered the D-Day address made by his great-grandfather, George VI, and met veterans during a ceremony at the National Memorial Arboretum in Alrewas, Staffordshire.

Meanwhile, in London, the Duke of Sussex joined six Normandy landings veterans at the Chelsea Pensioners’ annual Founder’s Day Parade.

D-Day on June 6 1944 was an the largest amphibious invasion of unprecedented scale and ultimately led to the liberation of Europe from Nazi occupation. At the start of the service, a message was read out on behalf of Pope Francis, in which he said D-Day was “decisive in the fight against Nazi barbarism”. 

He also paid tribute to those who “joined the Army and gave their lives for freedom and peace”.

Ms Sturgeon said: “It is a great honour to represent the people of Scotland at the commemorations marking the 75th anniversary of the D-Day landings. Scotland owes a great debt of gratitude to all those who served during the Second World War, especially to those who made the ultimate sacrifice.

“It is important that current and future generations continue to learn of the events that took place on this day 75 years ago, so that we can strive to ensure that such conflict is never repeated.

“It is also important the generations to come understand and recognise that it is the actions of those who served that allow us to enjoy the freedoms we now take for granted.”

Labour leader Jeremy Corbyn said the service was “beautiful and inclusive”, in memory of those “who died in Normandy and ultimately helped to defeat the scourge of fascism”.

The Prince of Wales said the day was “probably the last chance to pay everlasting respect” to the “remarkable” D-Day veterans.

He said: “I remember the last time we were there, I will never forget the determination that they still exude by being there. Being at the cemeteries and being at the commemorations. 

“They always remember their comrades and their friends and the people they had to leave behind. This country lost countless special people and we owe it to them and their families to go on commemorating this.”

Vladimir Putin brushed off his lack of an invite to the commemorations, saying he had “a lot business to do” in Russia. 

Speaking in St Petersburg as other world leaders gathered in Normandy to mark the occasion, he said: “We do not invite everyone to every event, so why should I be necessarily invited somewhere?

“Am I a person to be present at every event? I have a lot business to do here in Russia. But if we are imagining conspiracy theories, maybe western leaders wanted to have their own chit chat before having some contact with us.

“That doesn’t really matter. What matters is the truth about the history and about this horrible tragedy that happened to humanity during the Second World War. This is something we should always remember.”

Other politicians have also paid tribute to those involved in D-Day.

Scottish Conservative leader Ruth Davidson tweeted: “75 years ago 156,115 young men huddled onto landing craft, arrived in Normandy.

“They waded through sea and blood and fallen comrades. Their courage secured our freedom today.”

Scottish Labour leader Richard Leonard tweeted: “Today is a day for reflection on the sacrifice and the bravery of those who liberated France and so Europe from the iron heel of Nazi occupation” while LibDem leader Willie Rennie tweeted: “We owe them so much.”