Israel says its aim is to end the rocket attacks by Hamas-linked militants - of which there were reportedly more than 40 on Monday.
Defence Minister Ehud Barak said Israel wanted to deal Hamas a "severe blow" and its operation would be "widened and deepened as needed".
The US - Israel's strongest ally - said the onus was on Hamas to end the violence and commit itself to a truce, but there have been angry protests against the offensive in many cities across the Arab world and in several European capitals.
The strikes began less than a week after the expiry of a six-month-long ceasefire deal with Hamas, which has controlled Gaza since 2007.
Israel withdrew in 2005 but has kept tight control over access in and out of Gaza and its airspace.
GAZA VIOLENCE 27-29 DECEMBER
1. Ashdod: First attack so far north, Sunday 2. Ashkelon: One man killed, several injured in rocket attack, Monday 3. Sderot: rocket attacks 4. Nevitot: One man killed, several injured in rocket attack, Saturday 5. Civilian family reported killed in attack on Yabna refugee camp, Sunday 6. Israeli warplanes strike tunnels under Gaza/Egypt border, Sunday 7. Three young brothers reported killed in attack on Rafah, Sunday 8. Khan Younis: Four members of Islamic Jihad and a child reported killed, Sunday 9. Deir al-Balah: Palestinians injured, houses and buildings destroyed, Sunday 10. Interior ministry and Islamic University badly damaged, Monday 11. Gaza City port: naval vessels targeted, Sunday 12. Shati refugee camp: Home of Hamas leader Ismail Hanniyeh targeted, Monday 13. Intelligence building attacked, Sunday 14. Jebaliya refugee camp: several people killed in attack on mosque, Sunday
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.