Eid al-Fitr — the festival of breaking the fast
Eid al-Fitr falls on the 1st of Shawwal, the day immediately after Ramadan ends, celebrating the completion of a month of fasting.
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Islam has two annual celebrations — Eid al-Fitr, marking the end of Ramadan, and Eid al-Adha, marking the completion of Hajj and commemorating Ibrahim's sacrifice.
Eid al-Fitr falls on the 1st of Shawwal, the day immediately after Ramadan ends, celebrating the completion of a month of fasting.
كَانَ النَّبِيُّ صلى الله عليه وسلم لاَ يَغْدُو يَوْمَ الْفِطْرِ حَتَّى يَأْكُلَ تَمَرَاتٍ
"The Prophet ﷺ never went out on the Day of Fitr until he had eaten some dates."
Sahih al-Bukhari 986
أَمَرَنَا أَنْ نُخْرِجَهُنَّ فِي الْفِطْرِ وَالأَضْحَى الْعَوَاتِقَ وَالْحُيَّضَ وَذَوَاتِ الْخُدُورِ
"We were ordered [by the Prophet ﷺ] to bring out on Eid al-Fitr and Eid al-Adha even the young virgin girls staying in seclusion and menstruating women, [though menstruating women were to keep away from the actual prayer place]."
Sahih al-Bukhari 971
Eid al-Adha falls on the 10th of Dhul-Hijjah, during the days of Hajj, and commemorates Ibrahim's willingness to sacrifice his son in obedience to Allah — a test resolved when Allah provided a ram as ransom in his place.
وَفَدَيْنَـٰهُ بِذِبْحٍ عَظِيمٍۢ
"And We ransomed him with a great sacrifice."
See the full account in the story of Ibrahim. In remembrance of this, Muslims who are able perform Udhiyah — the ritual sacrifice of a sheep, goat, cow, or camel — distributing the meat between family, friends, and the poor. Unlike Eid al-Fitr, the sunnah on Eid al-Adha is to eat nothing before the prayer, then eat from the sacrifice afterward.
مَنْ ذَبَحَ قَبْلَ الصَّلاَةِ فَإِنَّمَا ذَبَحَ لِنَفْسِهِ، وَمَنْ ذَبَحَ بَعْدَ الصَّلاَةِ فَقَدْ تَمَّ نُسُكُهُ وَأَصَابَ سُنَّةَ الْمُسْلِمِينَ
"Whoever sacrifices before the prayer, he has sacrificed only for himself; but whoever sacrifices after the prayer, his sacrifice is complete and he has followed the way of the Muslims."
Sahih al-Bukhari 5556
Eid al-Adha also opens the three days of Tashriq (11th-13th Dhul-Hijjah), during which fasting is forbidden and the takbir is recited after every obligatory prayer — see the Practice page for the forbidden fasting days.
The Eid prayer is offered in congregation, without an adhan or iqamah beforehand — a detail explicitly preserved by the companions.
أَنَّ النَّبِيَّ صلى الله عليه وسلم صَلَّى يَوْمَ الْفِطْرِ رَكْعَتَيْنِ لَمْ يُصَلِّ قَبْلَهَا وَلاَ بَعْدَهَا، ثُمَّ أَتَى النِّسَاءَ وَمَعَهُ بِلاَلٌ
"The Prophet ﷺ prayed two rak'ahs on the Day of Fitr, and prayed nothing before or after it. Then he went towards the women along with Bilal..."
Sahih al-Bukhari 964
The prayer itself is two rak'ahs, with extra takbirs — additional declarations of "Allahu Akbar" — added before the recitation in each rak'ah (commonly seven in the first, five in the second, though the exact count is a point where the schools of fiqh differ slightly). It's followed by a khutbah, which — unlike the Friday khutbah — comes after the prayer rather than before it.
Recited on the way to the prayer and, on Eid al-Adha, after every obligatory prayer through the days of Tashriq:
اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ لَا إِلَٰهَ إِلَّا اللَّهُ وَاللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ وَلِلَّهِ الْحَمْدُ
"Allah is greatest, Allah is greatest, there is no god but Allah, Allah is greatest, Allah is greatest, and to Allah belongs all praise."