In one incident, thousands of Palestinian supporters from Syria entered the Golan Heights, Israel says.
Palestinians are marking the Nakba or Catastrophe, their term for the founding of the Israeli state in 1948.
Hundreds of thousands of Palestinians fled or were forced out of their homes in fighting after its creation.
Responding in a televised address to Sunday's violence, Israeli Prime Minister Benjamin Netanyahu said he hoped "calm and quiet will quickly return, but let nobody be mistaken, we are determined to defend our borders and sovereignty".
Clashes have been taking place at four separate borders or crossing points - at Erez in Gaza, near Ramallah in the West Bank, on the Golan Heights and at the border with Lebanon
Our correspondent, at the Qalandiya checkpoint there, says there is a stand-off now, but dozens of Palestinians have been injured.
Palestinian protesters have been throwing stones at Israeli security forces, who have been firing tear gas and rubber bullets.
On the occupied Golan Heights, the Israeli military said it had only fired warning shots as a large number of protesters tried to breach a border fence near the village of Majdal Shams.
But reports said at least two people had been killed and dozens injured.
Israel's army says this is a "serious" incursion. Brig Gen Yoav Mordechai said soldiers were still trying to control the crowds and that dozens of protesters had crossed.
The army has reportedly sealed off Majdal Shams and is carrying out house-to-house searches for "infiltrators".
Israel seized the strategic territory from Syria in the closing stages of the 1967 Six-Day War.
On the Lebanon-Israel border, a large number of protesters also approached the crossing with Israel.
Dozens of buses had brought protesters to the area under the rally slogan of "March for the return to Palestine".
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