The BBC's Ben Brown said there have been clashes between pro and anti Morsi demonstrators on the fringes of Tahrir Square
Egypt in crisis
Egypt's failed democratic experiment
Twelve people have died in Alexandria and three in Cairo in clashes between supporters and opponents of the ousted President Mohammed Morsi, reports say.
The violence came after three pro-Morsi demonstrators were killed by security forces in another part of the capital.
Troops later restored calm in Cairo, but nationwide violence left some 26 dead and 318 injured, officials said.
The army removed Mr Morsi from power on Wednesday after millions of people protested over his leadership.
Mr Morsi, Egypt's first freely elected leader, is in detention, as are some senior figures in the Brotherhood: early on Saturday, state media reported the Brotherhood's deputy leader Khairat el-Shater had been arrested at his Cairo home on suspicion of incitement to violence.
The Tamarod [Rebel] movement - which organised recent anti-Morsi protests - accused the ousted president of pursuing an Islamist agenda against the wishes most Egyptians, and of failing to tackle economic problems.
Anger and passion
Most of those killed during fighting in the Mediterranean city of Alexandria, died from gunshot wounds, said Amr Nasr, head of emergency services in the city.
He told the official Mena news agency that 200 people were injured during clashes in Egypt's second-largest city.
Earlier, after midday Prayers, Islamist supporters of Mr Morsi staged a series of marches across Cairo - including outside Rabaa al-Adawiya Mosque where tens of thousands massed.
Aleem MaqboolBBC News, Cairo
The scenario many had feared materialised on Friday evening. There were running battles in Cairo between those who oppose Mohammed Morsi and those who support him.
Through the day the Muslim Brotherhood crowds that gathered across the city had been largely peaceful. We followed protesters as they marched to the Republican Guard Officers' Club.
Earlier in the day pro-Morsi demonstrators had been killed there, but by late afternoon things had settled into a stand-off between soldiers and thousands of protesters.
But a separate crowd of Morsi supporters decided to march on the state TV building. We were with them as they stood only a few hundred metres away from the anti-Morsi crowds in Tahrir Square.
It was almost inevitable that clashes between the two groups would ensue. This was the first big security test for the army since their takeover of the country - and, after two hours, troops restored calm in the capital.
Tensions escalated when a crowd advanced on the nearby headquarters of the Republican Guard, where Mr Morsi is believed to be held.
Troops then opened fire on crowds. Three people were killed and dozens wounded, including the BBC's Jeremy Bowen whose head was grazed by shotgun pellets.
In the evening, tens of thousands of supporters of the Muslim Brotherhood - to whom Mr Morsi belongs - filled the square near the mosque, as well as nearby streets.
The Brotherhood's supreme leader, Mohammed Badie, told the crowd: "We shall stay in the squares until we bring President Morsi back to power."
He said their protests would remain peaceful and called on the army not to "direct your arms against us".
Shortly afterwards, Brotherhood supporters surged across the 6th October Bridge over the Nile river, towards Tahrir Square where anti-Morsi protesters were gathered.
The rival groups hurled fireworks and stones at each other. A car was set on fire and stones and fireworks were thrown.
The BBC's Kevin Connolly in Cairo says there is anger and passion on both sides - as well as a determination to win a battle for the streets which is making the capital a dangerous and volatile place.
Late on Friday, tanks arrived at the bridge to separate the clashing protesters and the violence died down.
There were clashes in other parts of Egypt on Friday.
Jeremy Bowen witnessed the shooting, and was also hit by shotgun pellets above the ear
In Qina in the south, troops opened fire on pro-Morsi activists trying to storm a security building. At least two people were injured.
Firing was also reported in the canal city of Ismailiya.
Ahead of Friday's protests, the army command said it would not take "arbitrary measures against any faction or political current" and would guarantee the right to protest, as long as demonstrations did not threaten national security.
"Peaceful protest and freedom of expression are rights guaranteed to everyone, which Egyptians have earned as one of the most important gains of their glorious revolution," it said.
On Thursday the head of Egypt's constitutional court, Adly Mahmud Mansour, was sworn in as interim head of state, and he promised to hold elections soon.
On Friday Mr Mansour dissolved the upper house - or Shura Council - which had been dominated by Morsi supporters and had served as sole legislative body after the lower house was dissolved last year.
Mr Mansour also appointed a new intelligence chief, Mohamed Ahmed Farid.