The complaint has been filed at a London court by ISPs Riseup and May First/People Link of the US, GreenNet of Britain, Greenhost of the Netherlands, Mango of Zimbabwe, Jinbonet of South Korea and the Chaos Computer Club of Germany, plus campaigners Privacy International.
They claim that GCHQ carried out "targeted operations against Internet service providers to conduct mass and intrusive surveillance."
The move follows a series of reports by German magazine Der Spiegel which claimed to detail GCHQ's illicit activities.
These reportedly included targeting a Belgian telecommunications company, Belgacom, where staff computers were infected with malware in a "quantum insert" attack to secure access to customers.
The legal complaint says this was "not an isolated attack" and alleges violations of Britain's Human Rights Act and the European Convention of Human Rights.
"These widespread attacks on providers and collectives undermine the trust we all place on the Internet and greatly endangers the world's most powerful tool for democracy and free expression," said Eric King, Privacy International's deputy director.
Britain's Foreign Office did not immediately comment.
GCHQ, which stands for Government Communications Headquarters, employs around 5,500 people and is housed in a giant doughnut-shaped building in the sleepy town of Cheltenham, southwest England.
Snowden's leaks claimed that the NSA had been secretly funding GCHQ to the tune of £100 million ($160 million, 120 million euros) over the last three years.