World leaders have welcomed a historic move by the US to end more than 50 years of hostility towards Cuba and restore diplomatic relations.
Pope Francis joined leaders from Latin America and Europe in praising the "historic" deal which saw the release of prisoners from both countries.
But dozens of dissident Cubans oppose the move, which some Republicans have labelled a "retreat" by the US.
US-Cuban ties have been frozen since the early 1960s.
President Obama said the "rigid and outdated policy" of isolating Cuba since then had clearly failed and that it was time for a new approach.
Mr Castro, meanwhile, has urged the US to ends its trade embargo, which has been in place since Cuba turned to communism more than 50 years ago.
But power to lift the embargo, which Mr Castro says has caused "enormous human and economic damage", lies with the US Congress, and correspondents say many Republicans are still deeply opposed to this.
'Beginning of the end'
Leading the praise, Pope Francis sent "warm congratulations" to Mr Obama and Cuban President Raul Castro for overcoming "the difficulties which have marked their recent history".
The announcement followed more than a year of secret talks in Canada and at the Vatican, directly involving the pontiff.
The European Union, which is in the process of normalising ties with Cuba, described the move as a "historical turning point", while leaders meeting at a Latin America summit in Argentina broke into applause at the news.
Chilean Foreign Minister Heraldo Munoz hailed it as "the beginning of the end of the Cold War in the Americas".
Venezuelan President Nicolas Maduro, whose predecessor Hugo Chavez was a close ally of Fidel Castro, said it was a "moral victory" and "victory for Fidel".
Former US Secretary of State Hillary Clinton said increased US engagement in Cuba in the future should "encourage real and lasting reforms for the Cuban people".
"And the other nations of the Americas should join us in this effort," she added.
Analysis: Julia Sweig, director for Latin America Studies at the Council on Foreign Relations
US President Barack Obama's re-election in 2012 and garnering of a sizable number of Cuban voters in Florida helped push the lifting of historic restrictions against Cuba.
It also increased the momentum to free US aid worker Alan Gross, following five years in a Cuban prison.
The added negotiations for the release of three Cubans in exchange for one US intelligence operative were the "most delicate", involving the highest levels of both governments, including Mr Obama and Cuba's Raul Castro.
But, as both countries now move towards regulatory efforts to liberalise trade and travel among them, negotiations will shift to lower level agencies.
Mr Obama's aim in using his executive authority to open up decades-long boundaries with Cuba is in "laying scaffolding" for longer-term political gains.
As more Americans travel to and trade with Cuba, pressure on Congress will mount to fully nullify the legally codified embargo. But I think we are a while away from that.
The Canadian prime minister, Stephen Harper, whose country never broke off ties with Cuba, welcomed what he called the "overdue development".
But the move was not applauded by everyone, with dozens of Cubans living in exile in the US state of Florida protesting after the announcement on Wednesday.
"It is a betrayal. The talks are only going to benefit Cuba," Carlos Munoz Fontanil said at a protest in Miami's Calle Ocho.
Meanwhile, Republican Senator Marco Rubio slammed the deal as "inexplicable", adding that it did nothing to address the issues of Cuba's political system and human rights record.
Cuban bloggers' reaction
"There were people who cried and those who did not believe what was happening... This change was something long-awaited by those who were born in the midst of a longstanding war that we did not ask for or want." - Alejandro Rodriguez
"Most people reacted with joy and hope. The coming days will show the real scope of the measures, but for now the streets exude optimism. Contrary to what uninformed people may believe, Cubans do not harbour anti-American sentiments."- Yuris Norido
"There were signs and indications that this big story did not come out of nowhere... Now it is time for civil society to deepen the struggle to establish a genuine state with a proper rule of law." - Regina Coyula
As part of the deal, US contractor Alan Gross, 65, was released from Cuban prison in return for three Cubans held in the US. President Obama said the US was looking to open an embassy in Havana in the coming months.
Officials said that Mr Obama and Mr Castro spoke by telephone on Tuesday for nearly an hour - the first presidential-level talks between the two nations since Cuba's 1959 revolution.
In exchange for Mr Gross, who was in poor health, and an unnamed intelligence officer, Washington released three members of the so-called "Cuban Five" who were serving lengthy sentences for espionage.
Mr Gross's five-year imprisonment had undermined previous attempts to thaw diplomatic relations between the two countries.
1959: Fidel Castro and his guerrilla army defeat the US-backed Cuban regime of Fulgencio Batista
1960-1961: Cuba nationalises US businesses without compensation; US breaks off diplomatic relations and imposes a trade embargo in response
1961: Failed Bay of Pigs invasion by CIA-backed Cuban exiles
1962: Soviet Union deploys ballistic missiles to Cuba, prompting Cuban Missile Crisis
2001: Five Cubans, dubbed the Cuban Five, are jailed in Miami for spying
2008: Raul Castro becomes Cuban president
2009: US citizen Alan Gross detained in Cuba accused of spying
Dec 2013: US President Barack Obama and Raul Castro shake hands at Nelson Mandela's funeral - the first such public gesture since 1959
17 December 2014: Alan Gross is released by Cuba