Syria's former president Bashar al-Assad is in Moscow with his family after Russia granted them asylum on humanitarian grounds, a Kremlin source told Russian news agencies on Sunday.

Russia's Foreign Ministry said that Assad had left Syria and had given orders for a peaceful transfer of power, after rebel fighters raced into Damascus unopposed on Sunday, ending nearly six decades of his family's rule and a brutal 13-year civil war.

"Syrian President Assad of Syria and members of his family have arrived in Moscow. Russia has granted them asylum on humanitarian grounds," state media quoted the unnamed Kremlin source as saying.

Meanwhile, Syrians awakened on Monday to a hopeful if uncertain future, after rebels seized the capital Damascus, forcing Assad to flee the country.

The lightning advance of a militia alliance spearheaded by Hayat al-Tahrir al-Sham (HTS), a former al-Qaeda affiliate, marked one of the biggest turning points for the Middle East in generations.

Assad's fall wiped out a bastion from which Iran and Russia exercised influence across the Arab world.

Many in the international community sought to take stock of a new-look Middle East following the end of Assad's autocratic rule.

US President Joe Biden said Syria is in a period of risk and uncertainty, and it is the first time in years that neither Russia, Iran nor Hezbollah held an influential role there.

The rebels face a monumental task of rebuilding and running a country after a war that left hundreds of thousands dead, cities pounded to dust and an economy hollowed by global sanctions. Syria will need billions of dollars in aid.

"A new history, my brothers, is being written in the entire region after this great victory," said Ahmed al-Sharaa, better known as Abu Mohammed al-Golani, the head of HTS.

Speaking to a huge crowd on Sunday at Damascus' Umayyad Mosque, a place of enormous religious significance, Golani said with hard work Syria would be "a beacon for the Islamic nation."

HTS is still designated as a terrorist group by the US, Turkey and the United Nations, although it has spent years trying to soften its image to reassure international governments and minority groups within Syria. (Reuters)