According to some sunni sources Ibadi are among the Khawarij sect. Ibadi themselves disagree but apparently from their books they at least approve the taughts of the Kharijite sect (Khawarij).
They seem to follow a student of 'Abdullah ibn 'Abas (May Allah be pleased with them) who's called Jabir ibn Zayd جابر بن زيد in Fiqh matters. But some scholars don't count him among the Ibadi or Khawarij sect. He was a great muffassir and muhadith who was taught also by um al-M'umminyn 'Aisha (May Allah be pleased with her) and he was a faqih of the ra'y-school. So either they are khwarij or sympathize with them and have the Tabi'i Jabir ibn Zayd as a theological leader!
Historically they began in Basra in the early 8th Century and became more organized with the Leadership of Abu 'Ubaida Muslim ibn Abi Karima in the middle of the 8th Century. At this time they began to build up communities in Sind, Khorasan, Hadramawt, Sana, Mekka and Medina and Later Tripolis (Lybia)... and made the basis for the later first Ibadi Sultanat of Tahart (Tiaret in Algeria) which was ruled by the Rustamid Dynasty.
Ibadi consider other muslims as ahl-al-Qibla
Ibadi nowadays live in Oman (where they are a majority 70% under the muslims) and in minoritys in some regions of Algeria, Lybia, Tunesia and Sansibar...
About Theology of Ibadi:
Sources: Quran, Sunna, Ijma', Qiyas and Istidlal.
- They accept ahadith from Sunni sources and have also their own sources.
- They don't accept hadith al-aahad أحاديث الآحاد in 'Aqida matters.
- Ibadi say nobody will see Allah neither in donya nor in akhira
- About the Question is the Quran created: They have 2 Opinions: the Ibadi of the Maghreb said yes the Quran is created while those from Oman ended with the opinion that the Quran isn't created!
- In the Imama matter: Ibadi say the muslims allways need an Imam to follow it doesn't matter if he is ibadi or not or if he is from Quraish or not.
- They consider the book called Musnad ar-Rabi' ibn Habib مسند الربيع بن حبيب as the most sane (Sahih) book of hadith as ar-Rabi' ibn Habib al-Azdi الربيع بن حبيب الأزدي is basically their third theological imam (leader) as he was a student of Abu 'Obaydah Muslim ibn abi Karimah أبو عبيدة مسلم بن أبي كريمة the student of Jabir ibn Zayd and narrated the hadith of the later (who was a student of teh sahabah as mentioned earlier) via his teacher.
[Source for the last 3 Points Wikipedia Articles Arabic and English]
Some Ibadi Point of Views in Fiqh:
They don't accept wiping over socks (المسح على الخفين) and say prayer is only valid if one washes his feet.
They say: In the dohr and 'asr prayer one doesn't need to recite a surah or verses from the quran after reciting the fatiha in the 2 first raka'at!
They hold the opinion that acts like:
- Qunoot in prayers,
- raising the hands when saying takbeer,
- moving the forefinger during the Tashahhud,
- saying "Amin" loudly after reciting the fatiha,
- and adding "As-salatu khayrun min an-Nawm" in the Adhan of fajr/sobh
are not valid acts.
According to Ibadi qassr during traveling is fard: And for example if you went to an other country without the niyya to stay there you could according their madhab always pray qassr!
- When it comes to fasting: ibadi say that one should do ghusl if having intercourse before fajr so according this if you didn't do ghusl your sawm is invalid!
- Again according to their Opinion some sins (for example lying) could make your sawm invalid!
- If one has to catch up days of Ramadan according the Ibadi Sect one should do them the same way he broke fast in Ramadan for example: if you missed 3 days successively you have to make them up 3 days successively
- In Zakat Ibadi made Sheep equal to Camels when it comes to Zakat.
- And they only allow to give Zakat to people who are in a case of wilaya otherwise if they can't find such a person they could give it to other muslims.
- Ibadi don't allow people who committed the sin of zina to marry each other!
- You will also find differences in inheritance matters and punishments.
Note that Ibadi's historically had a good co-existence with followers of the (Sunni) Maliki school of fiqh.
And Allah knows best!