The Israel Air Force struck targets in the Gaza Strip on Monday evening, after a rocket barrage was fired at the Jerusalem area and repeated rockets were launched at southern Israel.
The army said it was mulling a further response to what it called "an unacceptable attack."
Palestinian reports said that at least 20 people were killed in the IAF strikes, including a field commander for the Hamas terror group that controls the Strip.
Nine children were reported among the dead. Gaza's Health Ministry did not provide a breakdown on the cause of deaths. At least seven members of one family, including three children, were killed in an explosion in northern Gaza whose origins were unknown.
The death toll made it one of the bloodiest days of fighting in several years.
The rocket fire began a little after 6pm, the deadline Hamas had given Israel a short time earlier to remove its forces from the Temple Mount and Sheikh Jarrah, two East Jerusalem locations at the center of recent violent clashes between Palestinian protesters and Israeli police.
Palestinian sources also said that a motorcyclist had been killed in an IAF strike, while the military said it had foiled rocket fire from a cell of three Hamas members in the northern Strip.
The IDF Home Front Command has issued a ban on schools opening in the communities surrounding the Gaza Strip on Tuesday as well as in some areas further north.
The army also called on Israelis to obey the instructions issued by the IDF and avoid entering areas that have been declared out of bounds due to the security situation.
The attacks from Gaza included the Islamic Jihad launching an anti-tank missile at a jeep close to the border.
The terrorist organization reported that it had scored a direct hit. While the IDF confirmed the attack, it said the missile fired from northern Gaza had hit a civilian vehicle and that the driver had been hospitalized with minor wounds.
According to the army, the man was standing outside the vehicle when the rocket hit, which apparently prevented him from sustaining more serious injuries.
The IDF called the missile attack a "very serious incident that will not pass unanswered."
It said that most of the rockets fired Monday evening had been intercepted or fallen in open areas, while the anti-tank missile was directed at an unpopulated hill south of the city of Sderot.
"Hamas will pay a price for its conduct and we will respond severely," the army said.
"Hamas will understand this after we have taken action in the Strip. One option is to carry out a widespread operation. We have prepared an extensive and well-ordered plan and we will not hesitate to implement it," it said.
"This is an unacceptable attack on the State of Israel. Even now, we are bolstering our forces for a defense mission by the Gaza Division.
"We have prepared for an extensive range of scenarios, reinforced our air defense and deployed forces to put out fires" caused by arson attacks from the Strip.
President Reuven Rivlin said in response to the renewed rocket fire that "those who will test our resilience, will discover a fist of steel."
"We will not let anyone disrupt our lives, we will not bow down our heads in the face of any danger," he said.