The Afghan government has called for a humanitarian ceasefire with the Taliban, as the first-ever peace talks between the two sides began in Qatar.
Abdullah Abdullah, who led the government's delegation, stressed that there was "no winner through war".
The Taliban did not mention a truce, reiterating instead that Afghanistan should be under Islamic law.
The US encouraged both sides to reach an agreement, telling them: "The entire world wants you to succeed".
Afghanistan has seen four decades of conflict, with tens of thousands of civilians killed.
The historic talks began on Saturday, one day after the 19th anniversary of the deadly 9/11 al-Qaeda attacks in New York, that led to the US beginning military operations in Afghanistan.
The conflict in Afghanistan has been the longest in US history.
Why are these talks so important?
These are the first direct talks between the Taliban and representatives of the Afghan government. The militants had until now refused to meet the government, calling them powerless and American "puppets".
The conflict is still continuing in Afghanistan, and the government says 12,000 civilians have been killed since February.
At Saturday's opening ceremony, the head of Afghanistan's peace council, Abdullah Abdullah, called for an immediate ceasefire, telling Reuters "one of the top most issues on the minds of the people is reduction in violence in a significant way".
He added that his delegation "represents a political system that is supported by millions of men and women from a diversity of cultural, social and ethnic backgrounds in our homeland", and that they hoped "to close the gates of war and pain forever".
Taliban leader Mullah Baradar Akhund, meanwhile, said he hoped negotiations would "move forward with patience".
He added he wanted Afghanistan to be "independent, united, and… have an Islamic system in which all tribes and ethnicities of the country find themselves without any discrimination".
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The United States reached its own deal with the Taliban in February, and described the talks as a "truly momentous" opportunity.
"I think everyone sitting here today knows that it took hard work and sacrifice to reach this moment," Secretary of State Mike Pompeo said.
"The entire world wants you to succeed, and is counting on you to succeed."
An emotional day for delegates
Many Afghans who've travelled here described this day as emotional.
Today was about symbolism and a start to end a war which has touched every part of Afghanistan, and every family.
Both sides say today's first discussions on procedural issues have gone much better than expected, but on many other questions there are deep differences – the timing of a ceasefire, the shape of the political system, the extent of personal freedoms.
But more than anything else, the biggest issue is how to end a war now described as the deadliest in the world.
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