At least 15 Palestinians, including two women and a child, were reportedly hurt in the strikes.
Hamas earlier said its members fired rockets to respond to “Zionist aggression”, after accusing Israel of killing five of its fighters.
Israel denied the claim, but said 1,500 reservists had been called up.
Tension has spiked in recent days over the murders of three young Israelis and a Palestinian teenager.
The Israeli military said its offensive was part of a campaign, targeting Hamas militants.
“#IDF has commenced Operation Protective Edge in #Gaza against #Hamas, in order to stop the terror #Israel’s citizens face on a daily basis,” a spokesperson for the Israeli Defense Forces said in a tweet.
The military also deployed more troops near the Gaza border, reports say. However, the Israeli cabinet stopped short of ordering a ground operation for now.
Towns within 40km (24 miles) of the enclave were instructed by the central authorities to close schools and summer camps.
The army said that more rockets were fired into Israel from Gaza.
Hamas said Israel targeted two houses and four training facilities used by the militants across Gaza.
Palestinian medics said 15 people were injured, including two women and a child, in the southern town of Khan Younis.
Hamas militants reportedly warned they would enlarge the radius of their targets if Israel continued with the air strikes.
A Hamas spokesman had earlier accused Israel of killing the five militants during Sunday’s air strikes and called it a “grave escalation”.
He promised Israel would “pay a tremendous price”.
But Israeli military spokesman Lt Col Peter Lerner denied the claims, saying the men had died on Sunday in a tunnel that had been bombarded by Israel on Thursday.
He said the militants went into the tunnel to assess the damage from the air strike and meddled with some explosives, which were apparently detonated accidentally.
Israel said that Hamas had launched dozens of rockets on southern and central towns on Monday.
“Seven rockets were intercepted over the (southern) city of Ashdod and five rockets were intercepted over Netivot,” an army statement said, according to the AFP news agency.
Rocket alarm sirens were heard in the Hof Ashkelon and Sha’ar Hanegev areas.
There were no reports of casualties in Israel.
Meanwhile, Israeli Prime Minister Benjamin Netanyahu has been attempting to reassure the family of murdered Palestinian teenager Mohammed Abu Khdair that the authorities will take action.
The prime minister’s office said Mr Netanyahu had met the youth’s father and offered his condolences.