The collapse, which saw a vast stretch of the A10 freeway tumble on to railway lines in the northern port city, came as the bridge was undergoing maintenance work, and the country's deputy transport minister warned the death toll could climb further.
Italy's Interior Minister Matteo Salvini said that "around 30" have been killed in the incident.
"Unfortunately there are around 30 dead and many injured in a serious condition," Salvini told reporters in the Sicilian city of Catania.
Rescuers scouring through the wreckage, strewn among shrubland and train tracks, said there were "dozens" of victims, while images from the scene showed an entire carriageway plunged to the ground.
Cars and trucks were tangled in the rubble and nearby buildings damaged by vast chunks of concrete, according to an AFP photographer at the scene.
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Angelo Borrelli, head of the Civil Protection service which manages disaster areas, told reporters in Rome that an additional 13 people had been injured.
Aerial footage showed more than 200 metres (650 feet) of the viaduct, known locally as the Morandi bridge, completely destroyed.
The Italian fire service said that the bridge, located in an industrial area in the west of the city, collapsed at around noon (1000 GMT).
"I'm following with the utmost apprehension what is happening in Genoa and what looks like it could be an immense tragedy," Transport and Infrastructure minister Danilo Toninelli said on Twitter.
A fire service spokesperson told AFP that the bridge had mostly fallen on rail tracks 100 metres below and that cars and trucks had fallen with the rubble.
"Firefighters are working together and teams of rescuers and police sniffer dogs have been mobilised," emergency services said on Twitter.
The cause of the disaster was not immediately clear, although weather services in the Liguria region where Genoa is situated had issued a storm warning Tuesday morning.
The national motorways body said on its website that "maintenance works were being carried out on the base of the viaduct", adding that a crane had been moved on site to assist the work.
Shares in Italian company Atlantia, which runs much of Italy's motorway network including the collapsed stretch of the A10, plunged on the Milan stock exchange after the news.
They were down 9.7 per cent at 22.48 euros in mid-afternoon trading, having earlier been temporarily suspended when their slide exceeded 10 percent.
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European Commission chief Jean-Claude Juncker in a statement expressed "my deepest sympathy and sincere condolences to the families and friends of those who have died, and to the Italian people."
Genoa, home to half a million people, is located between the sea and the mountains of northwestern Italy.
Its rugged terrain means that motorways that run through the city and the surrounding area are characterised by long viaducts and tunnels.
The Morandi viaduct, completed in 1967, overspans dozens of railway lines as well as an industrial zone housing several factories.
One factory, immediately next to one of the viaduct's support columns, was virtually empty on Tuesday due to a national holiday, and seems to have sustained minimal damage.
Tuesday's incident is the latest in a string of bridge collapses in Italy, a country prone to damage from seismic activity but where infrastructure generally is showing the effects of economic stagnation.
In March last year, a couple were killed when a motorway overpass collapsed on their car near Ancona on the country's Adriatic coast.
A pensioner died in October 2016 when his car was crushed by a collapsing bridge over a dual carriageway between Milan and Lecco.
That incident was blamed on bureaucratic bungling which led to a fatal delay in the bridge being closed after it was reported to be showing significant cracks.
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