Final answer:
After Muhammad's death, vast territories including the Middle East, North Africa, the Iberian Peninsula, and Central Asia fell under Muslim rule through the expansions by the Rashidun, Umayads and Abbasid Caliphates, respectively, with further influence by later Muslim dynasties.
Step-by-step explanation:
In the decades after Muhammad's death, the spread of Muslim rule was extensive and rapid. The initial phase of expansion was carried out by the Rashidun Caliphate, which was succeeded by the power of the Umayad Caliphate. The Umayads moved the capital to Damascus, and under their rule, the Islamic empire extended into North Africa, the Iberian Peninsula, and parts of Central and South Asia. Following them, the Abbasids took over and brought about the Islamic Golden Age, with the capital at Baghdad until the Mongol sack in 1258. Islamic governance and influence continued with other entities like the Mamluk and Ottoman dynasties, despite occasional political fragmentation.
The conquests following Muhammad's passing saw territories such as modern-day Pakistan, much of Central Asia, the Middle East, North Africa, and the Iberian Peninsula being brought under Muslim control. The cultural and economic prosperity during these times made the Islamic caliphates powerful and admired centers of learning and development.