Palestinian bid for statehood is expected to be supported heavily in a 193-member UNGA, despite severe opposition by the US and Israel.
The US said that the Palestinian bid for a non-member observer state at the UN would be a "mistake".
The US has termed as a "mistake" the Palestinian bid to become a non-member observer state at the UN.
It was the beginning of an assessment process that will pit the aid-dependent Palestinians against the US and Israel.
Israel and the United States oppose the move and consider it a step back from long-stalled peace talks.
Abbas` jubilant mood was matched by celebration of thousands of Palestinians who thronged around outdoor screens in town squares across the West Bank.
President Mahmoud Abbas is pushing ahead with plans to submit the Palestinian application to the UN Security Council.
A veto by US, one of the five permanent members, would still block approval even if most other members agree.
Diplomats are working feverishly to guide the Israel and Palestine back into direct negotiations.
The Palestinians are expected to make their bid later this month before the UN General Assembly.
Day in Pics: 29th August
August 29, 2014
Famous Ganesh Pandals in Mumbai