Through workshops and other events, participants will explore factors that make people vulnerable to the crime and discuss its impact on their lives and their communities.
They will also examine the economies of places where victims are recruited and where they end up.
Thompson, who chairs the Helen Bamber Foundation, a Britain-based group that helps rebuild the lives of victims of cruelty, told the story of a woman from Moldova.
Searching for a better life, she was lured to England by a local woman who she befriended and trusted.
Once she got to Britain, she was forced into prostitution.
Costa, head of the UN office on drugs and crime, told the forum's opening session: "Two hundred years after the end of the transatlantic slave trade, we have the obligation to fight a crime that has no place in the 21st century. Let's call it what it is: modern slavery."
According to UN estimates, some 2.5 million people are involved in forced labour at any given time as a result of trafficking, and every continent and type of economy is affected by the crime.
The UN says 161 countries are reported to be affected and the majority of victims are aged between 18 and 24.
An estimated 1.2 million children are trafficked each year.
Suzanne Mubarak, Egypt's first lady, in outlining her country's efforts to tackle the issue, said the problem was "growing in severity and magnitude" and called it a "complex, multidimensional and ever-increasing scourge".
During the opening session, the global coalition Stop the Traffik handed Costa a petition of more than 1.5 million signatures.
Antonio Maria Costa and Ricky Martin with the
petition of more than 1.5m signatures [AFP]
"This is a historic global declaration," Stop the Traffik's chairman, Steve Chalke, said in a statement.
"Millions of people are not only aware of this crime but committed to do something for the millions who are trafficked.''
Costa warned that efforts to fight the crime were inadequate and that the exact magnitude of the problem has yet to be nailed down.
While the moral imperative to stop human trafficking has found its way on to policy agendas, governments are still not doing all they should, Costa said.
At a news conference later, he said "benign neglect" by government officials and concerns about protecting national reputations were hampering the fight.
Estimated annual profits from the exploitation of trafficked, forced labour is about $31.6m, according to material provided by the UN.