At one march, attended by thousands of Arab Israelis from northern Israel, hundreds of protesters clashed with police, and several people were injured, police said.
Israeli celebrations meanwhile drew large crowds to military air shows, while military bases opened their doors to the public to show off some of their arsenal.
The BBC's Tim Franks in Jerusalem says Israel has changed dramatically since it was founded, with a population nearly 10 times larger, and a stronger economy and military than its founders could have dreamt of.
However he says it is also a place riven by uncertainty - over its unresolved conflicts with neighbouring Arab countries and the Palestinians, and over the question of whether its own religious, political and ethnic mosaic still fits together.
Israelis were entertained by parachute jumps
The state of Israel was proclaimed about six months after the United Nations General Assembly voted to partition what was then Palestine into Jewish and Arab states.
In the war that followed, some 700,000 Palestinians fled or were driven out of their homes.
The anniversary is calculated according to the Jewish lunar calendar.
Palestinians are due to mark the occasion, al-Nakba, on 15 May.
Hopes for peace
The 60 years of Israel's existence have been marked by conflict, both with its Arab neighbours, and with Palestinians living under occupation.
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