The date - 21 December 2012 - is, some believe, the end of the "long count" calendar of the Mayan civilisation.
In China, police have arrested hundreds of members of a Christian group who apparently believe the prophecy.
Last year, experts said a new reading of the calendar showed that it did not in fact predict the apocalypse.
Many believe the date instead marks the start of a new era in the calendar.
Among the believers of the "apocalypse" themselves, there are different accounts as to when exactly the world should end.
Some said it would end at midnight on Thursday, while others gave the deadline of just after 11:00 GMT on Friday. Both predictions have failed to materialise.
Another group of followers said they were waiting for Friday's dawn, but that has also come and gone.
The misconception about the "prophecy" is not the only myth circulating about the Maya, the BBC's Will Grant in Mexico reports.
Many believe they were wiped out shortly after the arrival of the Spanish conquistadors in the 16th Century, our correspondent says.
But, in fact, the descendants of the people who built Chichen Itza, a Unesco World Heritage Site, are still here.
They are the second-largest ethnic group in Mexico, with between 800,000 and a million native Mayan speakers.
The ancient Maya flourished in modern-day Mexico and parts of Central America between 250 and 900AD.
Hundreds of spiritualists gathered in the city of Merida in Mexico, about an hour-and-a-half from the Mayan ruins of Chichen Itza.
Many prepared white clothes and incense to meet the sunrise, which - they believe - would bring the birth of a new age.
"The galactic bridge has been established," spiritual leader Alberto Arribalzaga said earlier this week.
"The cosmos is going to take us to a higher level of vibration... where humanity is in glory, in joy," he added.
In neighbouring Guatemala, thousands congregated on the Mayan ruins of Tikal in the jungles.
One spot thought by some to be destined to escape the end of the world is the mountain of Bugarach in southern France.
However, those preparing for the end of the world were reported to be far outnumbered by journalists.
The Turkish town of Sirince, another site reputed to be safe from the end of the world, saw similar scenes on Thursday.
Hundreds of reporters were wandering around the town of 570 residents, AFP news agency reported.
However, hotels around the Rtanj mountain in Serbia, a site rumoured to have magical powers, were booked out for the big date.
"I do not really believe that the end of the world is coming, but it is nice to be here in case something unusual happens," Darko, a 28-year-old designer visiting from Belgrade, told AFP.
In China, police have arrested almost 1,000 members of a Christian group which has predicted that Friday will usher in three days of darkness.
The group, called Almighty God, apparently urged its members to overthrow communism.
State media terms Almighty God an "evil cult", the same description it applies to the banned Falun Gong group.
The belief has gained considerable popularity in China, where the film 2012 was a box office hit.
A farmer in Hebei province, Liu Qiyuan - not a follower of Almighty God - has built seven survival pods which can contain 14 people each.
The pods, made of fibreglass, float on water and can survive storms.
Mr Liu told AFP news agency: "If there really is some kind of apocalypse then you could say I've made a contribution to the survival of humanity."
To calm anxieties, police in Beijing have posted an online notice telling people that "the so-called end of the world is a rumour".