After the death of Muhammad, caliphate was formed, which spread to vast geographical areas. The spread of Islam was spurred by missionary activities, by the Imams, who, by mixing with the local population, preached Islam, the way of life, practice. Such early caliphates, along with Muslim trade, spread Islam predominantly in the Middle East, up to the Pacific and the Atlantic.
Muslim dynasties are very fast established. Scientists from the Muslim world founded large school and university centers, libraries, where they gathered knowledge from the whole ancient world, which contributed to the golden age of Islam. During the golden era of Islam, Islamic writing expanded and consolidated through the written word.
Also, many dynasties strives for territorial expansion and embarked on the conquest, where, normally, the conquest implies the expansion of Islam. It is known that North Africa was conquered by Muslim dynasties, which until then had been Christian
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We can tell, that, spread of Islam has been achieved through word of mouth, peaceful measures, through writings and through war.