The protests, coinciding with preliminary results in south Sudan's landmark independence referendum and a sixth day of a revolt in Egypt, saw violent clashes in Khartoum and at least 64 people arrested, according to the police.
At the Islamic University of Omdurman, Khartoum's twin city, around 1,000 demonstrators were confronted by riot police as they marched, shouting slogans against President Omar al-Bashir, an AFP reporter said.
"Ocampo, what you have said is right!" they chanted, referring to the chief prosecutor of the International Criminal Court, Luis Moreno-Ocampo, who has accused Bashir of genocide and war crimes in Darfur.
Protesters hurled rocks at police who retaliated with tear gas and batons. Student members of the ruling National Congress Party (NCP) sided with police in some of the clashes, according to witnesses.
In the northern city of El-Obeid, around 600 kilometres (370 miles) west of Khartoum, riot police also used tear gas to disperse demonstrators.
About 600 people held what was billed as a peaceful protest in the city centre, shouting slogans against Bashir's government and calling for change, one witness said, before riot police fired tear gas to disperse the crowd.
In central Khartoum, a group of youths gathered near the presidential palace, chanting: "We want change! No to the high price of goods!" before anti-riot police chased off the protesters and arrested five of them.
Nearby, at the medical faculty of Khartoum University, security officers tried to prevent some 300 student protesters from leaving the campus.
But they eventually forced their way out onto the street, shouting: "Revolution against dictatorship!"
Police and security officers attacked them with batons, arresting several and forcing the students back inside the campus.
The official SUNA news agency later announced that the president had dismissed the university director, Mustafa Idris al-Bashir.
By the evening, a large security contingent, including more than 20 police trucks, had deployed around the university compound, with students reporting that they had been fired at with tear gas as they sat in the garden.
Opposition leader Yassir Arman, who was President Bashir's main rival in elections last year, slammed the "aggression" of the Sudanese security forces and demanded an end to one-party rule in the north.
"The aggressive handling by the security forces, police and ruling party members of peaceful demonstrations by students and young people... and the severe beatings and arrests of men and women... will only lead to more turmoil," Arman said in a statement.
Security officials also arrested three journalists and barred more than a dozen from covering the protests, while soldiers held an AFP cameraman for two hours.
The demonstrations followed calls by a Facebook group for peaceful anti-government rallies across Sudan and came after nearly a week of popular revolt in neighbouring Egypt that has shaken the government of President Hosni Mubarak.
"What we have seen in Egypt has inspired the youth to move, and they have organised themselves through the Internet," Mubarak al-Fadl, a leader of the opposition Umma party, told AFP.
"They want to show their anger that the affairs in Sudan have led to the partition of the country and because the future of the north is uncertain due to the policies of the government," he said.
Fadl said the youth-led protests had the support of "all the opposition parties," and he pinned blame for the secession of the south squarely on Bashir "and his minority Islamist group."
The first complete preliminary results of this month's referendum on southern independence, announced on Sunday, confirmed a landslide vote for secession.
In contrast to the jubilation in the south, widespread economic and political discontent have provoked sporadic street protests in north Sudan, with security forces maintaining tight control in Khartoum.
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