They want the government to resign, political prisoners to be released and the deaths of protesters investigated.
Six people were killed and many wounded in the last week as security forces used deadly force to quell protests.
Demonstrators have re-established a protest camp in Manama's central Pearl Square after security forces withdrew.
Hundreds of people spent the night in the square, the focal point of the week's unrest.
A protest camp had been cleared on Thursday with the deaths of four people, but protesters flocked to the square on Saturday after the military left.
They were met by police who fired volleys of tear gas and shotgun rounds, wounding dozens of people, before suddenly withdrawing and leaving the square to the jubilant protesters.
The kingdom's crown prince, Sheikh Salman bin Hamad al-Khalifah, has been charged to lead talks. In his capacity as deputy commander-in-chief of the armed forces, he ordered the military to return to their barracks - one of the key demands of the opposition for any talks.
Opposition figures have also said they want political reforms that will lead to a constitutional monarchy.
But some protesters have also called on King Hamad bin Isa al-Khalifah to step down.
Human rights activists are demanding to know what became of 10 people who they say went missing last week.
The BBC understands that the crown prince has made contact with representatives of all of the country's political parties, including the main Shia opposition parties in the Sunni-ruled Gulf state.
"Fellow citizens of Bahrain," the crown prince said in an address on state television.
"I hope that we can join hands, work together and communicate with all political forces in the country. Join us to calm the situation so that we can call a day of mourning for our lost sons."
Speaking on CNN television, he apologised for the deaths.
"I think there is a lot of anger, a lot of sadness, and on that note I would like to extend my condolences to all of the families who lost loved ones and all of those who have been injured. We are terribly sorry and this is a terrible tragedy for our nation," he said.
He said protesters would be allowed to stay in the square, which has become the focal point of the anti-government demonstrations.
They seem prepared for a long stay, setting up a makeshift hospital and erecting tents.
Bahrain is one of several Arab countries to have experienced pro-democracy demonstrations since the fall of long-time Tunisian President Zine El Abidine Ben Ali in January. Egypt's Hosni Mubarak was forced from power on 11 February.
The majority Shia population in Bahrain have long said they are discriminated against when it comes to housing and government jobs. They have also been calling for greater political rights from the Sunni royal family.
But the protesters have been careful to describe their revolt as non-sectarian, chanting slogans such as: "There are no Sunnis or Shias, just Bahraini unity."