Muhammad's successors gradually expanded Islamic rule through an affiliation of Islamic kingdoms and sultanates called the Caliphate ("successors of Muhammad").
The Caliphate in 750 AD
At its zenith, the Caliphate had spread well beyond the Arabian Peninsula. It stretched to the west as far as Spain, south across Morocco and all of N. Africa, all of Saudi Arabia, extending throughout the Middle East and into what is today India. This region was several times the size of the Byzantine Empire (north of the Mediterranean), of the same period.
The Caliphate dominated for 300 years until its power began to wane around the year 940 and finally ended March 3, 1924, with the abolition of the Ottoman Caliphate (Empire).
Note: Today some Muslims, such as Iranian President Mahmoud Ahmadinejad, believe it is their duty to work to restore the Caliphate's domination of the world, fulfilling Allah's will that Islam, the true religion, supplant all the corrupt religions that developed since the time of Abraham.
The fulfillment of that global destiny, as dictated by Islamic teachings, is why Usama Bin Laden insists that the United States and other nations convert to Islam.