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Mr Guaidó declared himself “acting president” on Wednesday
Venezuelan leader Nicolas Maduro has responded with defiance after the US recognised opposition leader Juan Guaidó as interim president.
Mr Maduro said Venezuela was breaking off relations with the US and gave diplomats 72 hours to leave.
But the US said “former President Maduro” no longer had the authority.
Mr Guaidó and the US have urged the army to abandon Mr Maduro, but the defence minister has instead condemned Mr Guaidó.
Seven South American nations including Brazil, Colombia and Peru have followed the US in backing Mr Guaidó as the legitimate president.
Canada is also supporting him, while the EU called for new elections.
However Mexico and Cuba Mexico, Bolivia and Cuba have expressed support for Mr Maduro.
Wednesday’s dramatic events took place amid mass protests against President Nicolás Maduro, who has overseen years of economic freefall.
Hyperinflation, power cuts and shortages of basic items have driven millions of people out of Venezuela.
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Mr Maduro was sworn in for a second term earlier this month, after a vote last May that was marred by an opposition boycott and widespread claims of vote-rigging.
What happened on Wednesday?
Mr Maduro accused Washington of trying to govern Venezuela from afar and said the opposition was seeking to stage a coup.
“We’ve had enough interventionism, here we have dignity, damn it!” he said in a televised address from the presidential palace.
Earlier President Donald Trump said he recognised Mr Guaidó as interim president in an apparently coordinated move minutes after the 35-year-old declared himself acting leader.
Mr Guaidó, who is head of the National Assembly, told a cheering crowd in Caracas that the protests would continue “until Venezuela is liberated”.
“I swear to formally assume the national executive powers as acting president,” he said, while raising his right hand, and promised to lead a transitional government and hold free elections.
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NGOs said 14 people had been shot dead during protests
He called on the armed forces – who have so far backed Mr Maduro – to disobey the government, but Venezuela’s defence minister accused him of conspiring with the US.
The BBC’s Latin America Editor Candace Piette says Mr Maduro has worked hard to keep the military leadership on his side, giving officers key government posts and offering lucrative oilfield services contracts to military-linked firms.
Venezuelan NGOs said that 14 people were shot dead during protests on Tuesday and Wednesday.
Wednesday also saw counter-demonstrations in support of Mr Maduro, but these were reported to be on a much smaller scale.
What did Trump say?
In a statement, he described Mr Maduro’s leadership as “illegitimate” and said the country’s congress was the only “legitimate branch of government” in the country.
“The people of Venezuela have courageously spoken out against Maduro and his regime and demanded freedom and the rule of law,” his statement said.
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The statement also said the US would hold Mr Maduro’s regime “directly responsible” for any threats to the safety of the Venezuelan people.
Mr Trump suggested tougher sanctions could be imposed on Venezuela.
He told journalists he was not considering military action but added that “all options are on the table”.
Secretary of State Mike Pompeo urged Venezuela’s military to support efforts to restore democracy and said the US would back Mr Guaidó in his attempts to establish a government.
Mr Trump also urged other nations to follow suit in supporting Mr Guaidó.
So far, Brazil, Colombia, Chile, Peru, Ecuador, Argentina and Paraguay have done so, along with Canada.
Mexico, Bolivia and Cuba however say they are backing Mr Maduro.
The Organization of American States (OAS) has also recognised Mr Guaidó as president.
In 2017 Venezuela announced it would withdraw from the organisation, which aims to aid co-operation across the continent – accusing it of meddling in its internal affairs.
Maduro’s biggest challenge
By Jonathan Marcus, BBC Diplomatic Correspondent
Several Latin American countries have followed the US lead, underscoring the regional antipathy to a regime that has compounded Venezuela’s many problems.
The question is what happens next? Does the Trump administration have a coherent plan for raising the pressure on the Maduro regime – freezing assets and so on? Crisis could simply lead to a greater calamity for the Venezuelan people.
Much will depend on which way the Venezuelan military jumps. For now, its generals may be backing the current regime. But will the lower ranks remain loyal to Mr Maduro or will they heed the growing unrest inside the country and the chorus of powerful voices coming from abroad?
Who is Juan Guaidó?
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Mr Guaidó has energised Venezuela’s opposition since he became head of the National Assembly
Mr Guaidó was a relatively unknown figure until he became president of Venezuela’s opposition-controlled parliament earlier this month.
Upon taking up the role, he said he had a constitutional right to assume the presidency until fresh elections were held.
As a student, he led protests against the late socialist leader Hugo Chavez who handpicked Mr Maduro as his successor.
His appointment as National Assembly leader has energised the opposition, which has been fragmented in recent years.
The opposition won parliamentary elections in 2015, but in 2017 Mr Maduro set up a separate body, the constituent assembly, which has taken over legislative powers.
Why are people protesting?
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Mr Maduro, who took office in 2013 after the death of Hugo Chavez, has been condemned at home and abroad for alleged human rights abuses and for his handling of the economy.
There are severe shortages of basic items such as medicine and food, and an estimated three million people have fled the country.
The annual inflation rate reached 1,300,000% in the 12 months to November 2018, according to a study by the National Assembly.
Wednesday’s demonstrations come just two days after 27 National Guard soldiers were reported to have revolted against the government at a guard post in the capital, Caracas.
But there is still a loyal core of people who support the government and say that Venezuela’s problems are caused by a right-wing opposition supported by the US and hostile neighbours.
They say that US sanctions have hampered the government by making it hard to restructure its debt.