UN Secretary General Kofi Annan has urged the world to make sure evils such as those perpetrated in the Holocaust are never repeated.
Six million Jews - and others - died at the hands of the Nazis
"We rightly say, 'Never again'. But action is much harder," he told the UN.
The General Assembly gathered to mark the approaching 60th anniversary of the liberation of the Auschwitz death camp in a special session.
German Chancellor Gerhard Schroeder will deliver a commemorative speech in Berlin on Tuesday.
Mr Annan said the United Nations, which was founded shortly after the end of World War II, must do everything in its power to prevent the scale of the slaughter undertaken by Nazi Germany.
"Two thirds of all Europe's Jews, including 1.5 million children, were murdered. An entire civilisation, which had contributed far beyond its numbers to the cultural and intellectual riches of Europe and the world, was uprooted, destroyed, laid waste," Mr Annan said.
"Truly it has been said: 'all that is needed for evil to triumph is that good men do nothing'."
He noted that Roma (Gypsies), Poles and other Slavs, Soviet prisoners of war, and mentally and physically disabled people "were likewise massacred in cold blood".
"Groups as disparate as Jehovah's Witnesses and homosexuals, as well as political opponents and many writers and artists, were treated with appalling brutality," he said.
But Mr Annan admitted that taking action against genocide was much more difficult than repeating rhetoric.
"The world has, to its shame, failed more than once to prevent or halt genocide - for instance in Cambodia, in Rwanda, and in the former Yugoslavia," he said.
He said "terrible things" were happening now in Sudan's western region of Darfur, and action was needed.
On Tuesday, Mr Annan is due to receive the findings of an international panel investigating the atrocities in Darfur.
Once the Security Council has the report, it will have to decide how to act, Mr Annan said.
Nobel Peace Prize laureate Elie Wiesel, himself a Holocaust survivor, also addressed the meeting in New York.
Mr Wiesel, whom Mr Annan described as a "dear friend", said he feared the lessons of Auschwitz had already been lost.
"If the world had listened we may have prevented Darfur, Cambodia, Bosnia and naturally Rwanda," Mr Wiesel told the 191-nation assembly.
"We know that for the dead it is too late... but it is not too late for today's children, ours and yours."
Israeli Foreign Minister Silvan Shalom said no-one would ever know if the UN could have prevented the Holocaust.
But he said the UN and each member state needed "to re-dedicate ourselves to ensuring that it will never happen again".
German Foreign Minister Joschka Fischer called the Holocaust "barbaric".
"For my country it signifies the absolute moral abomination, a denial of all things civilised without precedent or parallel," he told the assembly.
A worldwide week of events to mark the anniversary of the liberation of Auschwitz began on Monday in the Austrian capital, Vienna, with UN officials honouring the victims.