The £570m project to extend the Chilwell and Clifton lines began in March 2012.
The first public trams started running at 06:00 BST.
Despite facing criticism for causing disruption to businesses and commuters, Nottingham City Council insists the service will transform the local economy.
Delays were initially blamed on unexpected numbers of pipes and cables having to be moved, and then on work to replace part of the track bed.
£570m
Initial budget
28 new stations
11 miles of new track
22 new trams
20  million passenger capacity
Contractor Vinci has reported the problems contributed to a £165m loss for its civil engineering division.
Jane Urquhart, who has lead responsibility for operation company NET at Nottingham City Council, said it was a "really great day" for the city.
"With trams running from every seven minutes, it is all set to provide easier access to key locations right across the city," she said.
"The tram provides the backbone to the city's integrated transport network, underpinning all the efforts to reduce congestion, improve the environment and make Nottingham an even better place to live, work or visit."
NET and the city council had resisted calls to name a date for the new lines opening but final testing had been ongoing in recent weeks.
Steve Barber, vice president of the Light Rail Transit Association and former Broxtowe borough councillor said: "We are very relieved, as, I think, are a lot of people in Beeston.
"It has been a quite difficult ride over the past few years while construction has been at its worst ... and it is going to do wonders for the town and the city."
Stephen Knight, a butcher in Clifton, said: "We are optimistic. It has caused disruption, normal trade dropped by a third but the workers have helped.
"Now we will see if the public come back - but there are still a lot of parking restrictions outside."
Lisa Withers, a nearby florist, said: "I was really pleased to have been on the first tram from Clifton - and seeing the one from Toton arrive has been fantastic.
"It is a really great day for the city and I think we will now see some real regeneration and the tram driving Nottingham forward for people who live here and who work here and people who want to visit."
Full services on Nottingham's new tram lines have started, eight months behind schedule.