Although the Romans were successful in the first two Punic Wars, as they vied for dominance with the seafaring Phoenician city-state of Carthage in North Africa (modern day Tunisia), they did suffer a number of humiliations and damaging reverses. This built into an attitude of seeking vengeance and total victory that was expressed with these phrases
The attitude of total warfare toward Carthage resulted in the utter destruction of the city at the end of the Third Punic War. The city was ploughed over and surviving inhabitants sold into slavery. Historians dispute whether the fields were sown with salt, but the very notion is indicative of the vengeance wrought.
The term is sometimes adapted in modern usage, in a learned reference to total warfare, and has been used as the title for Alan Wilkins's 2007 play on the Third Punic War.
Although no ancient source gives the phrase exactly as it is usually quoted in modern times (either Carthago delenda est or the fuller Ceterum censeo Carthaginem esse delendam), something like this wording can be inferred from several ancient sources, which state that the Roman statesman Cato the Elder would always end his speeches with some variation of this expression even if he had not been discussing Carthage in the speech.