Barrow was sworn in neighbouring Senegal, holding a Koran as he took the oath of office at the Gambian embassy.
The oath was administered by the Gambian Bar Association president, Sheriff Tambadou, in the presence of ambassadors from countries in the UN Security Council and key African states.
There is little doubt that Barrow, who won last month’s election, will now be recognised internationally as The Gambia’s new head of state.
However, it remains unclear how soon he will be able to take power in Banjul.
West African regional grouping ECOWAS has mandated Senegal, which almost surrounds The Gambia, to lead a military intervention to install Barrow and its troops have massed on the border. Nigeria has sent aircraft and troops to Senegal, and deployed a warship.
However, West African military sources say intervention is intended only as a last resort ifJammeh cannot be persuaded by the threat of force to stand aside. “If no political solution is found, we will step in,” Col Abdou Ndiaye, a spokesman for the Senegalese military, was quoted by Reuters news agency as saying.
Gambia is a popular tourist destination and after the British Foreign Office updated its travel advice, holiday companies have been evacuating tourists.
Would military intervention be legal?
Senegal has circulated a draft resolution at the UN Security Council, seeking authorisation for an intervention but no vote has been scheduled. Some diplomats have suggested that such authorisation may not be necessary if Barrow, having assumed the presidency, requests help.
What is Jammeh’s position?
Jammeh tried to bolster his position earlier this week by declaring a state of emergency and engineering a parliamentary vote to extend his presidency by three months. He said these moves would prevent a power vacuum while the Supreme Court considered a legal challenge he has submitted over the election result.
Apparently disregarding the possibility that the court could eventually rule against him, he also promised to stay in power until new elections can be held.
Although the parliamentary vote took place while Jammeh was still officially president, regional powers are not regarding the decision as valid and are continuing with their plans to install Barrow at State House. Without this extension, Jammeh’s term ended at midnight on Thursday.
Would military intervention succeed?
A Senegalese-led intervention force would be expected to have more firepower than the Gambian army but it is not clear how much resistance it would face.
The army chief is a close Jammeh loyalist but has been quoted as saying he would not risk his men’s lives in a political dispute.
The elite regiment of the army, drawn mainly from the same ethnic group as Jammeh, is well-armed and is regarded as being fiercely loyal. However, it would be at a considerable numerical disadvantage to an intervening force.
The loyalty of other regiments is much less certain. Correspondents say there has been talk that some are unhappy with recent events and want a peaceful resolution.
How are the Gambian people reacting?
Tension is high in the Gambian capital, Banjul, over concerns that the political conflict will escalate into violence. Thousands of people have fled to neighbouring countries and rural areas.
Earlier this week, Barrow’s coalition urged Gambians to “exercise restraint, observe the rule of law and not to respond to provocation”.
What is the dispute over the election?
Jammeh, who seized power in a 1994 coup, initially accepted that Barrow had won the election but later reversed his position and said he would not step down.
He filed a petition to the Supreme Court, challenging the election results, and says the existence of this legal process means it would be unconstitutional for Barrow to assume office.
He also asked the Supreme Court for an injunction to stop the inauguration but the chief justice declined to rule on it, saying he must recuse himself from any case that could affect his own position – he would normally conduct the swearing-in ceremony.
Jammeh said there were irregularities in the election process, including the turning away of some of his supporters from polling stations, and errors made by the electoral commission.
The commission accepted that some of the results it initially published contained errors, but said Barrow still won.
Retaining power would ensure he was not prosecuted in The Gambia for alleged abuses committed during his rule.
How have his allies reacted to the crisis?
Ministers are continuing to desert Jammeh’s government. Several ministers, and even the long-serving vice-president, have resigned in recent days.
One of the ministers who resigned, former information chief Sheriff Bojang, said Jammeh’s arguments had a “veneer of constitutionalism” but were an attempt to subvert the will of the Gambian people. The international community says Barrow won fairly.
The lawyer representing Jammeh and his APRC party in the election challenge has also fled to Senegal and urged the long-time leader to step aside.
What is happening at the Supreme Court?
Judges from Nigeria and Sierra Leone have been hired to hear the petition to overturn the election result, but they have yet to arrive in Banjul.
Gambian chief justice Emmanuel Fagbenle has said the court will be able to convene no earlier than May, and possibly not until November, because the Nigerian who is to act as court president, Onogeme Uduma, is fully booked for the coming months.
How might this be resolved?
Barrow has distanced himself from comments, made by other opposition figures, suggesting Jammeh may be prosecuted over alleged abuses in power.
He said Jammeh should be able to stay in The Gambia and would be honoured as, and receive the privileges of, a former head of state if he stepped down.
However, analysts say Jammeh is unlikely to be persuaded by his opponent’s apparently conciliatory language.
Another option, raised by the House of Representatives is that Jammeh could be offered asylum and a comfortable retirement in another African country.
Besides Nigeria, Morocco has been mooted as a possible destination (Jammeh’s wife is Moroccan).
A third possibility has become more likely with Jammeh’s recent moves to entrench himself – that Jammeh continues to cling to power with the backing of at least some of the army, pending military intervention by regional powers.
December 1, 2016: Gambians go to the polls to choose their new president from three candidates, including Jammeh, who has ruled with an iron fist for 22 years, and Barrow, representing a large opposition coalition.
December 2, 2016: The electoral commission announces that Barrow has won, to scenes of jubilation on the streets. Jammeh concedes defeat.
December 5: Gambian opposition leader Ousainou Darboe, who was jailed in July for taking part in a protest, is freed on bail with 18 others. In the following days some 40 members of the opposition are also freed.
December 6: The electoral commission says Barrow won the election more narrowly than originally thought, and with a lower turnout. He garnered 43.2 percent of the vote and Jammeh 39.6 percent. A third candidate, Mama Kandeh, took 17.1 percent.
December 9: Jammeh announces he has changed his mind and says he rejects the results “in totality”. He lodges a complaint with the Supreme Court to overturn the result. The defiance sparks global concern, with the UN Security Council among those demanding that Jammeh hand over power.
December 13: Heads of state from the Economic Community of West African States (ECOWAS) are sent to The Gambia, but are unable to convince Jammeh to acknowledge his defeat.
December 31, Jammeh accuses ECOWAS of declaring “war”.
January 4, 2017: The army chief reaffirms his loyalty to Jammeh.
January 9: Jammeh dismisses 12 ambassadors after they call for him to step aside.
January 10: The Gambia’s chief Supreme Court justice says Jammeh’s legal challenge against the result will not be heard for several months.
January 13: The African Union (AU) urges Jammeh to respect the results and relinquish power peacefully, warning of “serious consequences”. It says it will not recognise him as of January 19.
January 15: Senegalese President Macky Sall announces he has agreed to take in Barrow until his inauguration.
January 16: The Gambia’s top judge pulls out of hearing a bid by Jammeh to halt the inauguration.
January 17: Jammeh declares a state of emergency, saying foreign powers had created an “unwarranted hostile atmosphere, threatening the sovereignty, peace, security and stability of the country”.
January 18: The Senegalese army says it is ready to intervene if there is no solution to the crisis. Gambian vice president Isatou Njie-Saidy resigns along with the environment and higher education ministers, the latest cabinet members to desert Jammeh. Nigeria and Ghana also agree to send troops. Mauritanian President Mohamed Ould Abdel Aziz travels to Banjul in an 11th-hour mediation bid.
January 19: A midnight (0000 GMT) deadline for Jammeh to step down expires. Barrow announces he will take the oath in the Gambian embassy in Senegal at 1600 GMT. UN Security Council to vote on a draft resolution endorsing the ECOWAS effort to ensure a transfer of power in The Gambia.