People queued at the city's Gyle shopping centre as the long-awaited first tram opened its doors to the public at 5am on Saturday.
The landmark day followed 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 Edinburgh has seen the construction of a line from Edinburgh Airport to York Place, costing around £776 million.
A tweet this morning from Edinburgh Trams said: "We sold 40,000 tram tickets over weekend. Likely to be lots more journeys too when returns and day tickets taken into account."
About 25,000 tickets were bought on Saturday and 15,000 on Sunday.
The transport system was praised by one of the very first paying passengers on Saturday.
Marjory Broom, her husband George and their son Christopher were excited to get on board.
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.''
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.
Many Edinburgh residents grew increasingly frustrated with the project as they had to endure roadworks and traffic congestion, while others remain angry at the scheme's over-budget cost.
As the project encountered delays and cost overruns, the other lines fell away, leaving the single route linking the city centre and the airport, but a review to be published later this year could lead to the extension of the tram system under the original plans.