"As ever, the most vulnerable continue to be on the frontlines of hardship and abuse."
He urged "those who are spared the most negative effects of disaster, poverty or instability" to not "turn a blind eye" to rights abuses.
"The cascading effects of abuse and indifference can eventually engulf the entire planet."
'Much to do'
On the eve of the 10 December anniversary, hundreds of Chinese lawyers, writers, academics and artists issued an online call for greater freedoms in China and democratic reforms, including an end to Communist one-party rule.
Two of the activists who signed the document, Zhang Zhuhua and Liu Xiaobo, were detained by police and questioned after the document - Charter 08 - was posted online, Reuters news agency said.
The UN High Commissioner for Human Rights, Navanethem Pillay, welcomed US President-elect Barack Obama's pledge to close the the US military prison at Guantanamo Bay, which houses hundreds of terrorism suspects detained since the 9/11 attacks.
UN High Commissioner for Human Rights Navanethem Pillay on DR Congo
She called for an inquiry into the facility, "so that conditions such as arbitrary detention, torture, cruel and inhuman treatment, are not ever again espoused by the US or copied by other countries".
Former UN High Commissioner for Human Rights, Mary Robinson, called the Declaration the "single most important global statement of the inherent dignity and equal rights of all people".
It "has inspired millions fighting for freedom and justice over the past six decades and continues to be a beacon of hope for people around the world," she told the BBC News website.
But she said much progress remained to be done towards fulfilling the ambitions set down in the Declaration.
She said that for women around the world, domestic violence remained a daily reality and that about one billion people "are daily denied basic rights to adequate food and clean water".
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