Security had been tightened in Cairo and other cities after Islamists called for protests against the government of President Abdel Fattah al-Sisi, the former army chief who ousted his Islamist predecessor Mohamed Morsi in 2013.
Demonstrators, mostly Islamist backers of Morsi, clashed with police, leaving 13 protesters dead in Cairo and another in the Mediterranean city of Alexandria, a health ministry official said.
Officials said the protester in Alexandria was shot dead by police after he opened fire on them.
Hundreds of other protesters who denounced both Islamists and the government also clashed with police in central Cairo.
The interior ministry said protesters shot dead a police conscript in the clashes in two north Cairo neighbourhoods that are strongholds of Islamist protests. Three other officers were wounded.
Elsewhere in Cairo, two policemen were wounded in a driveby shooting, security officials said.
At least 150 people were arrested across the country as police dispersed protests in which many leftwing demonstrators also participated, security officials said.
In downtown Cairo, police fired shotguns and tear gas against hundreds of protesters who tried to march on the central Tahrir Square, the epicentre of the early 2011 revolt that ousted Mubarak.
Armoured vehicles were stationed around Tahrir, but despite extra security in the capital jihadists set off a bomb in Cairo that wounded two policemen.
The interior ministry said the bomb exploded in eastern Cairo’s Alf Maskan neighbourhood, where a similar blast on Friday wounded four policemen and a civilian.