In a small room with two chairs side by side, Castro smiled as Obama said they would relay their concerns about each other's policies but could disagree with a spirit of respect.
"Over time, it is possible for us to turn the page and develop a new relationship between our two countries," Obama said. He said their immediate task was to reopen embassies in Havana and Washington.
After the two shook hands, Castro said he agreed with everything Obama had said. "We are willing to discuss everything, but we need to be patient, very patient," Castro said before reporters were ushered from the room.
In a later news conference, Obama said he was "optimistic that we'll continue to make progress, and that this can indeed be a turning point".
The meeting, on the sidelines of the 35-nation Summit of the Americas being held in Panama City, lasted roughly an hour, and the two leaders did not set a date for the re-establishment of diplomatic relations or the reopening of embassies in their capitals.
In a sign of the difficulty for the two nations in making basic, concrete progress towards overcoming their longstanding enmity, Cuba's Foreign Minister, Bruno Rodriguez, said the two countries would meet again for talks "as soon as possible", but were still working to achieve the "appropriate context" for formal ties.
The two leaders announced in December that they would normalise relations and that they intended to re-establish formal diplomatic ties. The opening of embassies in Washington and Havana has awaited the US removal of Cuba from its list of state sponsors of terrorism.
Obama, at his news conference, said the recommendation to remove Cuba from the list, which Havana has said must precede the opening of embassies, was now on his desk. "I've been on the road," he said. "I want to make sure I have a chance to read it and study it before we announce it publicly." White House officials said the announcement would be made "in the coming days".
"The concerns around the embassy are going to be mostly on the Cuban side," Obama said. "They haven't dealt with an American Embassy in Cuba for quite some time."
US negotiators have said Cuba must give US diplomats freedom to travel around the island, and allow Cuban citizens free access to the embassy.
Calling his talk with Castro "candid and fruitful", Obama said, "I can tell you that in the conversations I've had so far with him - two on the phone and most recently face to face - we are able to speak honestly about our differences ... we have very different views on how society should be organised." He added that the US was "not going to stop talking about human rights, free expression and freedom of the press".
Obama's welcome of Cuba ended a longstanding irritant with Latin America, most of which mended fences with Cuba decades ago, and added a potential legacy issue to his presidency alongside a nuclear deal with Iran, assuming he is able to achieve it.
Throughout his four-day swing through the region, which began on Thursday with a stop in Jamaica, Obama has been upbeat and perhaps relieved to deal with regional relations that, even including Cuba, are far less fraught than the constant drumbeat of death and destruction in the Middle East, and sparring with Russia over Ukraine.
While there are opponents to his Cuba outreach in Congress, they are not nearly as numerous, or dangerous, as those in both parties who have questioned aspects of his foreign policy.
Reflecting the other problems he faces, Obama said: "Cuba is not a threat to the United States. It doesn't mean we don't have differences, but on the list of threats that I'm concerned about, I think it's fair to say that between [Isis], Iran getting a nuclear weapon, activities in Yemen and Libya ... Russian aggression in Ukraine ... climate change, I could go down a pretty long list."
Despite Castro's offer to discuss "any issue," the Cuban leader also told Obama his Government would not accept US efforts to "bring about changes in the political and economic system in our country", according to Rodriguez.
Their meeting followed a dramatic speech to the gathering in which Castro called Obama "an honest man ... a humble man", who was not to blame for what he said was more than a century of US oppression of Cuba and who was "courageously" trying to persuade Congress to lift its embargo against his country.
Speaking for nearly an hour, Castro first pounded the table and recounted the history of what he called US oppression and abuse of Cuba, from early military incursions and occupation to the Bay of Pigs invasion and what he described as US-sponsored terrorist attacks on the island.
"I apologise to Obama for expressing myself so emotionally," Castro said. "President Obama has no responsibility for this. There were 10 presidents before him; all have a debt to us, but not President Obama ... I have read his books - parts of them - and I admire his life." As he spoke, Obama sat without expression, his eyes downcast.
Cubans welcome historic meeting but are keen to see action soon
Cubans hailed yesterday's historic meeting between Presidents Raul Castro and Barack Obama but said they wanted to see ties warm faster between the Cold War foes so that the lives of ordinary people on the island could improve. Havana residents stopped in the middle of errands or family time to watch Castro and Obama shaking hands and addressing the press about their efforts to re-establish diplomatic ties.
"The fact that Raul and Obama sat down to talk in a civilised way after all these years of such serious tensions seems historic to me," said Roger Rodriguez, a university professor.
Irene Quintana, a homemaker, said she was cleaning house when her grandmother called her over to watch the meeting on television.
"It seems magnificent to me, incredible. It excited me so much and I think that it's hopeful," she said.
The Castro-Obama meeting was the most dramatic development since the two Presidents' announcement on December 17 that they would release captured spies and restore full diplomatic relations as part of an effort to broadly normalise relations between the two countries.
The announcement was greeted inside Cuba by jubilation but emotions have since cooled as the two countries have struggled to strike a deal on the first landmark in normalisation - the re-establishment of full embassies in Havana and Washington.
Many Cubans are eager to see embassies and, as soon as possible, an opening of greater trade and tourism from the US that will help Cuba's stagnant economy and the daily struggles of ordinary people, whose salaries of US$20 ($26.50) a month on average make it tough to put food on the table.
"I like that Raul left all the doors open. That seems important to me," said Magaly Delgado, a retired office worker. "We'll see if it leads to results."
Rosa Marie Argudin, a street performer, said: "It has been years that we've been waiting for something like this. I hope this doesn't just remain a conversation."
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