At least 20,000 people have died in the huge earthquake which struck south-eastern Iran, according to officials.
The earthquake reduced much of the city to rubble
Rescuers worked through the night to hunt for survivors from the quake, which struck before dawn on Friday.
The ancient city of Bam has been devastated - most of its buildings have been flattened including two hospitals and a 16th Century citadel.
A huge relief operation is under way, with many foreign countries sending supplies and rescue workers to Iran.
Thousands of people who lost their home in the quake spent the night outside in the bitter cold and into the early hours of Saturday.
President Mohammad Khatami described the quake as a "national tragedy" and said it was too huge for Iran to cope with alone.
The United States offered humanitarian assistance, and President George W Bush said he was "ready to help" Iran.
Aid agencies are making urgent appeals for supplies such as tents, blankets and field hospitals.
But there were fears for the safety of survivors as temperatures overnight in the desert city fell to below freezing, coupled with the shortages of food and water.
The country suffers frequent earthquakes, with small tremors happening almost daily. In one earthquake in 1990, 35,000 people died.
Bam and the surrounding area is home to more than 200,000 people. About 70% of the houses in Bam have been destroyed, Iranian state television reported.
It is thought many people were crushed as they slept.
There were scenes of intense grief in the city, with people weeping next to corpses shrouded in blankets.
"I have lost all my family. My parents, my grandmother and two sisters are under the rubble," Maryam, 17, told Reuters.
Two of the city's hospitals have collapsed, crushing many staff, and remaining hospitals are full. Many of the injured are having to be flown elsewhere for treatment.
A huge relief operation involving ordinary Iranians, the army, Islamic volunteer groups and local rescue teams is under way.
Rescue teams are flying to the area by plane and helicopter.
Emergency centres set up in makeshift buildings are now trying to care for the tens of thousands who have been injured.
RECENT IRAN QUAKES
June 2002: more than 200 killed in the western Qasvin and Hamedan regions
May 1997: More than 1,500 killed in eastern Iran
February 1997: about 1,000 killed in north-western Iran
June 1990: 35,000 died in worst recorded disaster in Iran, affecting the Caspian regions of Gilan
and Zanjan. About half a million people made homeless
June 1981: More than 1,000 killed in quake that destroyed town of Golbaf
The BBC's Miranda Eeles in Tehran says that in many places there simply is not enough room.
Telephones, electricity and water supplies have been cut by the tremor.
The concern now is how to rescue those still alive but buried deep under the rubble, our correspondent adds.
A number of countries - including Russia, Britain, Germany and Spain - have offered to send relief teams and supplies.
The United Nations said it was sending experts to co-ordinate the aid effort.
The Red Crescent is setting up tents to house survivors.
Friday's quake had a magnitude of at least 6.3, according to Iranian sources. The US Geological Survey measured it at 6.7.
Bam - about 1,000 kilometres (620 miles) south-east of Tehran - was on the Unesco's list of World Heritage Sites.
An important regional centre in the 16th and 17th centuries, it contained many ancient buildings that were not built to withstand earthquakes.
Since 1991, tremors have claimed some 17,600 lives and injured 53,000 people, according to official figures.