But the US and several other nuclear powers say a ban won't work and the world should instead stick with a more gradual approach.
"There is nothing I want more for my family than a world with no nuclear weapons. But we have to be realistic. Is there anyone who thinks that North Korea would ban nuclear weapons?" The New York Times reported U.S Ambassador Nikki R. Haley as saying to reports outside the General Assembly as the talks were getting underway.
Even Japan — the only country to have suffered atomic attacks, in 1945 -- voted against the talks, saying the lack of consensus over the negotiations could undermine progress on effective nuclear disarmament.
The countries leading the effort include Austria, Ireland, Mexico, Brazil, South Africa and Sweden. Hundreds of NGOs back their efforts.
They say the threat of nuclear disaster is growing thanks to mounting tensions fanned by North Korea's nuclear weapons program and an unpredictable new administration in Washington.
Supporters point to successful grassroots movements that led to the prohibition of landmines in 1997 and cluster munitions in 2008.
"I expect that this will take a long time, let's not be naive," Swedish Foreign Minister Margot Wallstrom said at the UN last week.
"But it's very important in these days when you see more of this rhetoric, and also sort of power demonstrations, including threatening to use nuclear weapons."
"Quite a high number of countries are actually interested in saying we have to break the deadlock that has been on this issue for so many years," she added. "So it's also the expression of frustration."
No progress has been made on nuclear disarmament in recent years despite commitments made by the major nuclear powers to work toward disarmament under the 1968 Non-Proliferation Treaty (NPT), said Beatrice Fihn, director of the International
Campaign to Abolish Nuclear Weapons, an international
coalition of NGOs.
"There was disappointment with the Obama administration, which made some pledges, but then ignored most of them," she said. "And now there are raised worries with the new US president."
Then-president Barack Obama announced a drive in 2009 to reduce the role of nuclear weapons and eventually eliminate them.
But his administration strongly encouraged NATO allies to vote against this year's UN negotiations, saying a ban would obstruct cooperation to respond to nuclear threats from adversaries.
With Agency inputs