Around 20 people have died and hundreds detained.
President Kabila was supposed to step down on Tuesday and has blamed logistical problems for delaying his departure.
The unrest has sparked fears the region could again spiral into conflict.
More than a day after he was supposed to step down, Democratic Republic of the Congo president Joseph Kabila (kuh-BEE-luh) remains in the top job.
The leader has been in power for two five-year terms and, under the country's constitution, cannot run for a third.
Mr Kabila says logistical issues, such as difficulty in registering voters, have delayed holding elections to choose his successor.
And Information Minister Lambert Mende has denied it is a tactic to hold onto the presidency, saying the President will step down when required.
"He is just trying to respect the constitution. He must remain in office until a new president is elected. Why should I guarantee something that is written in the constitution? When people are asking us to guarantee, that is a way of insulting us. This is racism. We wrote the constitution, My Dear, we have to respect the constitution. When a foreign government comes and says to our head of state, 'You must guarantee that what you've sworn, that you have to respect the constitution ... (it) will be respected,' that is to abuse our president."
Protests broke out across the nation of 70 million people after the news of Mr Kabila staying on, before they appeared to diminish.
The final number of dead and wounded from the protests remains unclear, but police say close to 300 people were arrested.
Those dead include a United Nations peacekeeper from South Africa and a local police officer.
Human Rights Watch senior researcher Ida Sawyer says residents are angry at the President's actions.
"In addition to the protest and repression by the security forces, we're also seeing mobilisation by armed groups in many parts of the country. Over the past few weeks, armed-group leaders have been saying that, starting on December 20th, President Kabila would no longer be legitimate, the army and the police were, therefore, no longer legitimate, and they would take security into their own hands. We've already seen, since Monday, heavy fighting when a militia attacked in the town of Butembo. At least 13 were reported to be killed, in eastern Congo. And then yesterday, in the south, in Kananga in the Kasai area, a militia also attacked, and there was heavy fighting during the day yesterday between this militia and the army in south-eastern Congo. In Manono, a Batwa militia attacked, at least 40 people reportedly killed. All of this seems to be linked to President Kabila's continued attempts to extend his (stay) in power."
On Wednesday, some residents began trying to resume normal life, although traffic and public transport were noticeably reduced.
The country, Africa's biggest producer of copper and other metals such as cobalt, has struggled to be free of conflict since gaining independence from Belgium in 1960.
Its Catholic bishops have been acting as mediators in talks between the government and opposition groups.
Pope Francis has issued a plea to all parties to, as he put it, "be actors of reconciliation and of peace."
Felix Basse, from the United Nations peacekeeping mission in the country, MONUSCO, says the situation in the capital Kinshasa is close to breaking point.
"So far, in Kinshasa, it looks like most of the shops are closed, schools are not open. Some calls were made today by the president of the Rassemblement (party), calling on people to refrain from any violence, but to have a peaceful demonstration, to stage a peaceful resistance, to what they call the end of term of President Kabila. But what needs to be kept in mind is that things move very fast here, and we are calling on all the Congolese to demonstrate responsibility and to observe and respect the rules of the republic, but also to refrain from any inflammatory speech or any action that may create tension or create violence."
The Democratic Republic of the Congo's constitutional court has ruled Joseph Kabila can continue to lead until elections, scheduled for mid-2018, are held.
But the controversy has also affected the nation's diplomatic relationships.
Germany has announced it is postponing talks over development assistance, due to what it calls the "restricted" scope for action by the government.
