First Punic War 264 - 241 B.C.:
By the time of the first Punic War, Rome had expanded through Italy and was eager to control Magna Graecia. Carthage's involvement in the Greek area of southern Italy provided an opportunity and launched the first of 3 wars between rival Mediterranean powers.
Second Punic War 218 - 201 B.C.:
Although Rome eventually won the second Punic War, there were some tense moments as Carthage's skilled military leader, Hannibal, conquered Roman territory. As a side effect, Hannibal taught the Romans military tactics that they later used against him.
Third Punic War 149 - 146 B.C.:
By the time of the Third Punic War, Rome was more powerful than Carthage, but Carthage still represented an annoying threat, so Rome made sure Carthage wouldn't rise again. It is said that Rome salted the earth just to make sure.