Three thousand polling stations, staffed by thousands of PA-appointed civil servants, including teachers, opened at 7am (0500 GMT) on Sunday, but few voters showed up in the early hours, possibly due to cold weather.
According to the PA election committee, close to 1.8 million eligible voters could take part in the election in the West Bank, Israeli-occupied Arab East Jerusalem and the Gaza Strip.
The presidential election is the second since the PA was established under interim peace agreements with Israel. Arafat was elected in 1996.
Hamas and its junior Islamist sister, the Islamic Jihad, are boycotting the elections on grounds they are based on the 1993 Oslo Agreement between Israel and the PLO that the two groups rejected.
Hamas has argued that the elections are "tailored to suit only one candidate," referring to interim PLO leader and Fatah candidate Mahmud Abbas.
It is not clear whether the boycott will affect voter turnout.
Opinion polls and officials have suggested that turnout possibly will exceed 50% of eligible voters.
Some polls predicted that up to 80% of eligible voters would cast ballots.
A high turnout would be good news for Abbas, also known as Abu Mazin, who is widely expected to win the elections by a large margin.
Pre-election polls indicated Abbas would win the presidential election by a landslide after a crowd-pleasing campaign in which he pledged to uphold Arafat's struggle for statehood in Israeli-occupied territories, but by nonviolent means.
Al-Barghuthi is expected to
receive 20-25% of the votes
Abbas was forecast to take 52% to 65% of the vote, more than twice the support commanded by his closest challenger, human rights activist Mustafa al-Barghuthi.
Al-Barghuthi is expected to receive 20% to 25% of the votes.
An independent candidate, Sayid Baraka, is also expected to perform well, especially in the Gaza Strip.
However, it is unlikely that either of the two candidates will pose a serious challenge to Abbas, who is strongly backed by the Fatah movement.
Analysts said Abbas would probably need at least 60% of the vote and a large turnout to build a popular mandate for peace talks with Israel.
Early results of the elections are expected to be announced around midnight on Sunday, two hours after ballot boxes are closed.
The official results will be announced in Ram Allah on Monday.
The elections are being monitored by hundreds of observers from scores of countries.
Among the most prominent observers who have arrived in the occupied territories are former US president Jimmy Carter, former French premier Michele Rocard and EU Foreign Policy Chief Javier Solana.
Mustafa al-Barghuthi, the National Initiative
Bassam al-Salhi, the Palestinian People's Party
Taysir Khalid, the Democratic Front for the Liberation of Palestine
Sayid Hasan Baraka, independent
Abd al-Halim al-Ashqar, independent
Abd al-Karim Shbair, independent
Moreover, there is a conspicuous presence of international media, including representatives from Japan, China as well as Western news agencies.
In addition to monitoring the ballots and trying to ensure the transparency of the elections, the observers will monitor Israeli army interference with Palestinian freedom of movement.
The Israeli government had promised its occupation army, which tightly controls the West Bank and most of the Gaza Strip, would leave Palestinian population centres for 72 hours to allow a free and smooth election.
However, Israel said it may renege on this promise, citing unspecified security reasons.
On Sunday, the Israeli army announced that major roadblocks in the West Bank would remain despite the elections.
"Major checkpoints and roadblocks are still in place. There was never any intention to remove them. It is obvious they must stay in place for security reasons," an Israeli army spokesman was quoted as saying.
The roadblocks are expected to interfere with the freedom of movement of Palestinians and may discourage many from going to the polls.
Palestinian officials announced on Saturday that legislative elections in the occupied territories would take place on 17 July.
The announcement is expected to be welcomed by Hamas, which has been urging the PA to name a date for the elections.
Seven candidates are contesting and Mahmud Abbas, from the dominant Fatah movement, is tipped to win. His latest opinion poll rating was 65%.
In second place is human rights activist Mustafa al-Barghuthi who had 22% in the most recent poll. The other candidates are expected to get only a few per cent between them.
Voters in East Jerusalem, annexed by Israel in a move not recognised internationally, will cast ballots at Israeli post offices.
About 800 international observers, including former US president Jimmy Carter and former French prime minister Michel Rocard, will join 20,000 local monitors.
The resistance movement has indicated that it will participate in the parliamentary elections as well as in the remaining phases of the upcoming local elections.
Hamas participated in the 23 December municipal elections in 26 Palestinian towns, making significant achievements in what were thought to be traditional Fatah strongholds.
Despite the generally positive atmosphere surrounding the elections, Palestinians seem to have no illusions as to what these polls could mean in terms of liberating them form decades of military occupation by Israel.
"Even if we were to develop the best functioning democracy in the world, we still would not be free. Our main priority is not really democracy but ending the occupation, since no real democracy, or for that matter economic development, can possibly exist in the absence of freedom," said one Palestinian academic.
"Our problem is the occupation, the occupation, the occupation," said an elderly man who was on his way to cast his ballot in al-Khalil, in the southern West Bank.
Nonetheless, most Palestinians feel the elections will be a valuable asset toward political development, not only in Palestine, but throughout the Arab world.