Security forces fired teargas and automatic weapons into the air as demonstrators massed in Tahrir Square, Cairo, the focal point of the revolt.
State television said that 49 people were killed in clashes during protests in Cairo and elsewhere.
The core demands of the 2011 revolt - freedom and social justice - could only be heard in protests outside Tahrir Square.
However, crowds gathered in central Cairo after nightfall on Saturday to call for an end to the army-backed government, shouting: "Down with military rule!"
Armoured personnel carriers were deployed to restore order and anyone entering Tahrir Square had to pass through metal detectors.
A car bomb exploded near a police camp in Suez on Saturday, said security sources, with reports that nine people were injured. The blast was followed by fierce exchange of gunfire.
The demonstrations showed the popularity of General Abdel Fattah al-Sisi, whom many are calling on to run for president. The leader of the Coptic Christian church backed Sisi's military takeover.
The general, who served as head of military intelligence under Mubarak, is expected to announce his candidacy for the presidency soon, according to Reuters. He is likely to win by a landslide in elections, expected within six months.
"He's a good man. He's a hero. I want Sisi to be my president," a protester at Tahrir Square told BBC News.
But not everyone is happy with the turn of political events. "Basically, this isn't the third anniversary for the revolution that we were hoping for," said Egyptian writer Ahdaf Soueif. "The security state is back and also a great many activists are in jail."
An al-Qaida-inspired group based in the Sinai Peninsula claimed responsibility for the bomb that exploded near a police academy on Saturday, according to the SITE monitoring organisation.
In an audio message posted on militant websites, al-Qaida leader Ayman al-Zawahri called on Egyptian Muslims to focus on fighting what he called "an Americanised coup" staged by Sisi instead of battling the country's minority Christians.
Watch scenes from Tahrir Square on the third anniversary of the Egypt Revolution