Israel said the Iran-backed group had declared war on Saturday as its army confirmed fighting with militants in several Israeli towns and military bases near Gaza and Prime Minister Benjamin Netanyahu vowed to retaliate.
"Our enemy will pay a price the type of which it has never known," he said.
"We are in a war and we will win it."
At least 22 Israelis have been killed in the attack so far with more than 250 wounded, Israel's ambulance service said, adding the toll was expected to rise.
The Israeli military said it had launched air strikes into Gaza, where witnesses reported hearing heavy explosions, with at least two dead.
The attack marked an unprecedented infiltration by an unknown number of Hamas gunmen into Israel from Gaza and one of the most serious escalations in the Israeli-Palestinian conflict in years.
Palestinian militants in Gaza have fired thousands of rockets into Israel, which is retaliating.
Israeli broadcaster Reshet 13 TV News said militants were holding Israelis hostage in the town of Ofakim, and that five Palestinian militants had been killed in the town of Sderot, where homes had been set on fire.
Israeli media reported gun battles between bands of Palestinian fighters and security forces in towns in southern Israel.
Israel's police chief said there were "21 active scenes" in southern Israel, indicating the extent of the attack.
In Gaza, people rushed to buy supplies in anticipation of days of conflict ahead.
Some evacuated their homes and headed for shelters.
Hamas military commander Mohammad Deif announced the start of the operation in a broadcast on Hamas media, calling on Palestinians everywhere to fight.
"This is the day of the greatest battle to end the last occupation on earth," he said, adding that 5000 rockets had been launched.
The last major flare-up between Israel and Hamas was a 10-day war in 2021.
Hamas military commander Mohammad Deif has called on Palestinians everywhere to fight.
Israeli Defence Minister Gallant said "troops are fighting against the enemy at every location" and authorised the call-up of reservists.
The Israeli military said its forces were operating inside Gaza but gave no details.
Israeli media reported that gunmen had opened fire on passers-by in Sderot, and footage circulating on social media appeared to show clashes in city streets as well as gunmen in jeeps roaming the countryside.
The escalation comes against a backdrop of surging violence between Israel and Palestinian militants in the West Bank, which together with the Gaza Strip is part of the territories where Palestinians have long sought to establish a state.
It also comes at a time of political upheaval in Israel, which has been riven by deep divisions over moves to overhaul the judiciary, and as Washington works to strike a deal that would normalise ties between Israel and Saudi Arabia.
The Iran-backed Lebanese group Hezbollah said the operation was a "decisive response to Israel's continued occupation and a message to those seeking normalisation with Israel".
Palestinian media says Israelis have been taken captive by fighters and tanks have been destroyed.
Hamas media displayed videos of what it said were bodies of Israeli soldiers brought into Gaza by fighters, and Palestinian gunmen inside Israeli homes and touring an Israeli town in jeeps reportedly been driven into Israel by the attackers.
Reuters was not immediately able to verify the footage.
Palestinian media also reported that a number of Israelis had been taken captive by fighters and Hamas media circulated video footage apparently showing a destroyed Israeli tank.
The Israeli military was aware of reports of captives, a security source said but provided no further details.
In a briefing with reporters, an Israeli military spokesman declined to comment.
In Gaza, the roar of rocket launches could be heard and residents reported armed clashes along the separation fence with Israel, near the southern town of Khan Younis, and said they had seen significant movement of armed fighters.
The Palestinian Islamic Jihad group said its fighters were joining Hamas in the attack.
The attack came a day after Israel marked the 50th anniversary of the 1973 war that brought the country to the verge of catastrophic defeat in a surprise attack by Syria and Egypt.
