The UN hopes the attacks will galvanise global action
Delegates will hear an opening speech by New York Mayor Rudolph Giuliani, who has been widely praised for the way he has led the city following the devastating 11 September suicide attacks on the World Trade Center and the Pentagon.
UN Secretary General Kofi Annan will then urge members to sign and ratify 12 existing UN conventions on terrorism, and to adopt a new convention which would close many of the loopholes in existing international law.
The week-long meeting comes as Americans are being warned by their government that "further terrorist activity" is likely in response to prospective military action against Afghanistan.
The ruling Taleban in Afghanistan have been told to hand over the main suspect in the attacks, Saudi-born militant Osama Bin Laden, or face the consequences.
In other developments:
Taleban will 'pay price'
To counter the threat, US Attorney General John Ashcroft urged Congress to enact new anti-terrorism laws before the end of the week, and called for what he called a "national neighbourhood watch" to prevent further acts of terror.
More than 6,000 people lost their lives when four US civilian airliners were hijacked simultaneously on 11 September and flown into buildings housing thousands of workers.
On Sunday, the Taleban said that Bin Laden was still in Afghanistan at an unknown location.
US Defence Secretary Donald Rumsfeld said it was "self-evident" the Taleban would pay the price if they did not hand him over.
The United States has massed troops, warplanes and aircraft carriers within striking distance of Afghanistan in preparation for possible military action.
Some reports say US and UK special forces are already operating within Afghanistan.
Defining terrorism
Almost no country in the world has escaped untouched from terrorist acts, and delegates from UN member states at this week's meeting will be relating their experiences and their suggestions about how to deal with the problem.
The 12 existing UN conventions on terrorism were largely ignored before last month's atrocities.
Their ratification and the adoption of a new global convention proposed by India have so far been delayed by arguments over the definition of terrorism.
This centres on the old dispute that one person's terrorist may be another person's freedom fighter.
The BBC's UN correspondent Greg Barrow says there is hope that the gathering will give added momentum to the Indian initiative.
Indian diplomats say this problem can be resolved by defining what constitutes an act of terrorism as anything that goes beyond the Geneva Conventions on war.