An Israeli woman was stabbed and moderately wounded in one such attack, while a Palestinian allegedly tried to stab an Israeli soldier and was shot dead in another, the police and army said.
UN High Commissioner for Human Rights Zeid Ra'ad Al Hussein said the latest flare-up in violence in the six-decade-old conflict was "dangerous in the extreme".
"The violence between Palestinians and the Israelis will draw us ever closer to a catastrophe if not stopped immediately," he said during a meeting of the UN Human Rights Council in Geneva.
But Abbas said today that "it is no longer useful to waste time in negotiations" and warned that a continuation of the current violence could "kill the last shred of hope for the two-state-solution-based peace."
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He called on the United Nations "to set up a special regime for international protection for the Palestinian people."
Abbas accused Israel of "extrajudicial killings of defenceless Palestinian civilians, (and having) detained their corpses, including children."
Palestinians have long feared Israelis are planning to change the rules governing the site that is sacred to both Muslims and Jews, and lies in a majority Palestinian area annexed by Israel in 1967.
Israeli Prime Minister Benjamin Netanyahu has repeatedly denied seeking to allow Jews to pray at the compound, which they refer to as the Temple Mount.
Hundreds of angry Palestinians demonstrated last night in Hebron to demand the return of "the bodies of martyrs" -- youths behind the wave of unrest that has seen nine Israelis killed in knife attacks and shootings.
The West Bank city of Hebron has been a hotbed of the recent unrest, with near-daily clashes with Israeli police where protesters are often left with bullet wounds or dead.
Many of the youths behind the attacks are also from the city.