This 'hood isn't strictly a geographical, rather a time-related, cultural sphere.
Therefore it comprises a different area than Mesopotamia, linked together by the rule and strong influence of Islamic culture. It comprises Islam's cradle, modern day Saudi Arabia and adjacent countries, except Persia ("Arabia"), stretches all over the Levant ("Bilad ash-Shâm"), across northern Africa ("Maghreb") over to Spain ("al Andalus").
The first capital of the Caliphate, after Prophet Muhammad had died in 632, was Medina in modern day Saudi Arabia; by that time it stretched across the entire Arabian peninsula and into parts of what is now Syria and Iraq. Within a century an Islamic state spread out from the Atlantic ocean to Central Asia, bringing forth empires of the Umayyads, Abbasids, Mughals, Seljuq Turk, Safavid Persia and Ottomans, which all ranged among the most powerful in the world. The era of the Abbasids is also referred to as "The Golden Age of Islam".
Several Caliphs (Abu Bakr, Umar, Usman and Ali), called the Rashidun (rightly guided), followed after the death of Muhammad, before a series of Caliphates was established. A Caliphate was like a monarchy with its own laws that had adopted a particular Islamic sect as state religion. Until the 9th century the Muslim world presented itself as one single political entity under the leadership of one Caliph. The early Caliphate was also called the Arabic Empire or Islamic Empire and ended with the murder of Ali during the First Fitna (civil war) in 661.
Ali was succeeded by Mu'awyah I, the first hereditary Caliph of the Umayyads. During his reign his armies managed to conquer most of North Africa, the Caliphate of the Umayyads expanded under his successors into North Africa, the Maghreb and Iberia in the west and Central Asia in the east. The reign of the Umayyads was interrupted in the 680's by a series of revolts, the Second Fitna, but was re-established and lasted until 750.
It is said that the Abbasids were descendants of Abbas, the uncle of Muhammad, claiming that they were the saviours of the people under the Umayyad rule. Under the Abbasids, the expanse of Umayyad empire was consolidated and enhanced by the Mediterranean islands, including the Balearics and Sicily. In the Abbasid era the arts flourished, most notably prose and poetry. This age is also referred to as The Golden Age of Islam. The name of one Abbasid Caliph became immortal by a collection of stories, One Thousand and One Nights - Harun al-Rashid, the main figure and character throughout several of the oldest versions of 1001 Nights. Baghdad became the new capital of the Caliphate to emphasise the Abbasid's interest in the affairs of Persia and Transoxiania.
By then, the vast Abbasid empire began to crumble; one surviving family member of the Umayyads, Abd ar-Rahman I, had managed to escape to Iberia and had established an independent Caliphate there in 756. The Aghlabids, an Arabic family, had been appointed autonomous rulers of the Maghreb by Harun al-Rashid, though they still accepted the authority of the central Caliphate - at least for a while. Soon after, in 909, the Aghlabids were succeeded by the Shiite Fatimite who had conquered Abbasid Egypt by 960, built their new capital there in 973, called al-Cahirah (the planet of victory) - modern day Cairo. The remaining parts of the Abbasids empire were eventually consumed by the Seljuq Turks in 1055.
However, expansion continued (by warfare and/or peaceful proselytism): The first stage of the conquest of India, sub-Saharan West Africa. Under the Abbasids, Baghdad blossomed to one of the greatest cultural centres of the world. This era also encouraged the rise of classical Sufism, the Mystic Path of Islam. Both, Harun al-Rashid and al-Mamun, were generous patrons of arts and sciences which flourished considerably under their reigns.
Credits:
Alhambra, photo by Sir Cam, public domain (inscription: "help" and "God")
Harun al-Rashid, public domain
Maqmat, public domain
Map "Age of the Caliphs": US government, public domain
Trellis border and divider: SwissWebArt
Masthead background: Persian Garden, Sarmarcande Bahktiari's Kilim Bazaar
"Muslim History" and several other sources on the web
Text, background, masthead and layout: Alal-Sin Malachus