One of the attacks, which saw an assailant open fire on a bus, marked the first assault with a gun in Jerusalem in the two-week-old upsurge of Palestinian violence.
The rising tide of unrest, which has seen a series of stabbing attacks and violent protests, has raised fears that a full-scale third Palestinian uprising, or intifada, could erupt.
In the first and second intifadas, in 1987-1993 and 2000-2005, hundreds were killed in near-daily violence between Israel and the Palestinians.
Two Israelis were killed, including a 60-year-old, while around 10 others were wounded, medics and police said.
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One of the attackers was shot dead while the other was wounded, police said.
"I was in my house and I heard between 20 and 30 shots between the police and the terrorists," a man who lives in the area said, declining to give his name.
In another attack around the same time in an ultra-Orthodox Jewish neighbourhood of Jerusalem, an assailant drove a car into a group at a bus stop and exited with a knife.
It was not clear if the attacks were coordinated.
The Jerusalem attacks came after a Palestinian stabbed and wounded a passerby north of Tel Aviv before being overcome by civilians in the area.
Police described the attacker as a 22-year-old Palestinian from east Jerusalem and said he was taken to hospital with serious injuries after being beaten by residents.
After first breaking out with attacks and violent protests in annexed east Jerusalem and the occupied West Bank, unrest has also spread to the Gaza Strip, with clashes along the border in recent days leaving nine Palestinians dead from Israeli fire.
Funerals of Palestinians killed during unrest or while allegedly carrying out attacks -- as well as videos of the shootings -- have fed anger.
Palestine Liberation Organisation secretary general Saeb Erekat said today new petitions would be sent to the International Criminal Court against Israel, accusing it of "extrajudicial executions".