Prince Harry says he hopes the Invictus Games will "inspire people" affected by mental illness to seek help.
The prince, who founded the games for injured veterans, said at the opening ceremony in Florida that it was "not just physical injuries that our Invictus competitors have overcome".
This is the second time the games have been run, after London in 2014.
Athletes from 14 countries will compete over four days, with finals in rowing, powerlifting and archery on Monday.
Micky Yule won Britain's first Invictus Games gold medal in the men's lightweight powerlifting competition.
A former staff sergeant, he served with the Royal Engineers and lost both of his legs above the knee when he stepped on an improvised explosive device (IED) in Afghanistan in 2010.
The 39-year-old, from Musselburgh near Edinburgh, described Prince Harry as a "top lad".
"Do you know what, he's such a supporter and I think without Prince Harry's input into the whole games, I'm sure it wouldn't be what it is," he said.
"He puts himself out, he's around everybody. You see him in the back helping everybody."
Thousands of people attended the opening ceremony on Sunday, which included speeches from US First Lady Michelle Obama, Hollywood actor Morgan Freeman and former US President George W Bush. There were also performances by British singer James Blunt and the soprano Laura Wright.
Prince Harry paid tribute to the competitors during the opening ceremony.
"Every single one of them will have confronted tremendous emotional and mental challenges. When we give a standing ovation to the competitor with the missing limbs, let's also cheer our hearts out for the man who overcame anxiety so severe he couldn't leave his house.
"Let's cheer for the woman who fought through post-traumatic stress and let's celebrate the soldier who was brave enough to get help for his depression."
He added: "To those of you watching at home and who are suffering from mental illness in silence - whether a veteran or a civilian, a mum or a dad, a teenager or a grandparent - I hope you see the bravery of our Invictus champions who have confronted invisible injuries and I hope you are inspired to ask for the help that you need."
The BBC's Aleem Maqbool, who was at the ceremony, said it was "at some times a very emotional event with injured veterans telling their stories" and Prince Harry describing his own experience of 10 years in the British Army.
The prince also joined Mrs Obama in an interview for US television where he said he wanted to "smash the stigma" around mental health problems, adding that conditions such as depression and post-traumatic stress were "not a life sentence".
The US First Lady told Good Morning America the Games were a part of rehabilitation for many injured servicemen and women.
"Once they take that uniform off they are still looking for ways to serve and we can't waste that talent," she said.
Prince Harry also revealed how he was determined to accept the US President and First Lady's challenge and get the Queen involved in a video to promote the games after the couple had acted "so confident" about the home team during their recent visit to London.
He said: "I didn't want to ask the Queen because I didn't want to back her into a corner but when I showed her their video and told her, she said 'Right, what do we need to do? Let's do this.'"
British cycling competitor Craig Preece, who suffered injuries to both legs in a bomb blast in Afghanistan, said he was looking forward to being able "to showcase to the world what we can still do".
The Paralympic-style tournament will feature track and field athletics, indoor rowing, sitting volleyball, wheelchair rugby, tennis and basketball as well as cycling, swimming, archery and triathlon.
As well as powerlifting, Monday's schedule includes rowing and archery.
Live coverage, behind-the-scenes action and commentary will also be featured on the BBC Sport website.
Coverage of day one of the Games including highlights of the opening ceremony will be broadcast on BBC One at 19:30 and 20:30 BST each day.