Out of Mecca, East to India, West across North Africa, And on into Spain.
The Muslim Empire began circa A.D. 600, with the founding of Islam by Mohammed, and ended in A.D. 1258 with the Mongol conquest of Baghdad, the Muslim Capital. This period is known as the Muslim Golden Age.
The Muslim Empire encompassed much of the modern middle east, along with much of northern Africa and a portion of southern Spain. The Empire would have extended farther into Europe save for the efforts of the Franks. Under the leadership of Charles Martel, the Frankish armies routed the Muslim forces at the battle of Tours in A.D. 732. Why were the Muslims so effective in war, one might ask. The answer to this question is quite simple. The Muslims had been taught to convert the world to Islam, by force if necessary. The Muslims believed that were they to die in service to their religion, they would go directly to heaven. This belief made them fierce fighters, they won either way: were they to die, they would be in heaven; were they to live, they were alive and could take part in the spoils of war.
The Muslim Empire acted as the library of the Dark Ages. While most of the rest of the world was falling into barbarism and feudalism, the Muslims were studying and learning from the knowledge of previous civilizations such as the Greeks and Romans. The Muslims also collected knowledge from their neighbors: Chinese gunpowder, Indian mathematics, and Byzantine architecture were all spread through the Muslims.
In addition to spreading the ideas of their neighbors, the Muslims also contributed their own ideas. Those of us who hate Algebra and Trigonometry have the Muslims to thank for our misery. The Muslims also expanded the Greek knowledge of the stars and created accurate charts of the night skies. The Muslim medical system was centuries ahead of its neighbors. The Muslims had hospitals, textbooks for their doctors to learn from, and they even required their physicians to pass an exam in order to practice medicine. The Muslims also developed a banking system, they even implemented a credit system, in which citizens could borrow money and pay it back at a later date.
One immensely important contribution of the Muslim Empire was their preservation of Greek and Roman culture. They acquired elements of this from the Byzantine Empire, and not only did they protect the cultures of these great civilizations, they also added to them, translating texts, repairing works of art, etc.