And they have been hailed as "absolutely fantastic" by one of the travellers on the very first service.
Tram enthusiasts Marjory Broom, 59, and her husband George, 63, were on board as the first service left the city's Gyle shopping centre at 5am today
The landmark day for the project follows six years of building work and well-documented problems, including a long-running dispute between the council and its contractor.
Overall, the controversial scheme to return trams to the streets of Scotland's capital has seen the construction of a line from Edinburgh Airport to York Place, costing around £776 million.
The sight of trams running in Edinburgh will not be new to the city's residents, since testing of the system got under way earlier this year.
But today the new service - described as "an efficient and sustainable" addition to Edinburgh's transport options - was fully launched and opened to the public.
Mrs Broom, her husband and their son Christopher, 31, were excited to be on the first tram early this morning.
She said: "We're all tram enthusiasts so we've been following the progress of the trams for some time.
"I thought they were fantastic, absolutely fantastic. It was chock-a-block, and it was a real carnival atmosphere on board with people cheering as the tram set off."
She added "We've been on trams all over the world and the tram in Edinburgh today was really, really smooth. It was the first tram and it was standing room only, it was very, very busy.
"I'm really excited we managed to get on - there would have been no buzz being on the second tram."
The family all travelled on the first tram as it went from the Gyle shopping centre to York Place in Edinburgh city centre, before heading out to Edinburgh Airport and then back to the Gyle.
Mr Broom praised the new service, saying: "The Edinburgh trams, I think they are the best I've been on, they're very very wide and very, very smooth. I think they will be very succesful. We just have to hope they extend the service so it goes down to Leith and possibly even Granton, then it would be a proper network."
He added: "It was great to see the trams have finally arrived, it was good to go along Princes Street on the first tram, looking up at Edinburgh Castle with the sun shining."
Also on the first tram was Edinburgh City Council transport convener Lesley Hinds, who conceded the transport project had been "hugely challenging" but said there was now "growing positivity" about the trams.
"Two years ago, I pledged to do everything in my power to ensure we brought this project in on the revised budget and revised schedule," she said
"It was never going to be straightforward, but with an incredible team effort and a focus on results, it now feels fantastic to be where we are today.
"The tram project has been hugely challenging for the people of Edinburgh but now there seems to be a growing positivity in the city about the start of a brand new transport service.
"I know from first hand experience how trams can transform and enhance a city and I am determined Edinburgh should make the very most of the opportunities ahead."
The project sees the return of trams to the capital for the first time in around 58 years.
The new network was initially intended to be larger than it is now, stretching to the waterfront at Leith and Newhaven, and it was originally hoped to be in operation in 2010.
But as the project encountered delays and cost overruns, the other lines fell away, leaving the single route linking the city centre and the airport.
Many Edinburgh residents also grew increasingly frustrated with the project as they had to endure roadworks and traffic congestion, while others remain angry at the over-budget cost of the project.
Transport bosses are now keeping their fingers crossed that everyone who uses a tram in these first few days and weeks enjoys the experience.
Ian Craig, chief executive of Transport for Edinburgh, said: "Following many months of preparation for the new service, Transport for Edinburgh is now primed to play a central role in the future success of Edinburgh, moving residents and visitors to where they need to be on a dynamic and fully integrated transport system, utilising tram and bus."
He added: "I'm proud of what has been achieved but there's much more to be done. This is a brand new system and, as we start to welcome passengers on board, we'll be keeping a very close eye on the service to make sure our customers get the best possible experience."