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Carthage never recovered militarily after the Second Punic War, but quickly economically and the Third Punic War that followed was in reality a simple punitive mission after the neighbouring Numidians allied to Rome robbed/attacked Carthaginian merchants. Treaties had forbidden any war with Roman allies, and defense against robbing/pirates was considered as "war action": Rome decided to annihilate the city of Carthage. Carthage was almost defenceless, and submitted when besieged. However, the Romans demanded complete surrender and moval of the city into the (desert) inland far off any coastal or harbour region, and the Carthaginians refused. The city was besieged, stormed, and completely destroyed. Ultimately, all of Carthage's North African and Iberian territories were acquired by Rome. Note that "Carthage" was not an 'empire', but a league of punic colonies (port cities in the western mediterranean) like the 1st and 2nd Athenian ("attic") leagues, under leadership of Carthage. Punic Carthago was gone, but the other punic cities in the western mediterranean flourished under Roman rule.

What cities flourished after they were conquered by the Romans?
punic cities