In the first Punic War the Romans fought the Carthaginians on the water. The Romans were not trained as well as the Carthaginians, so the Roman made a bridge system on their boats so they could board the Carthaginians ships and use their strong army to take them out. The Romans won this war. The Romans got Sicily.
During the second Punic War Hannibal pushed his way to the gates of Rome beating army after army of Roman soldiers. He took elephants with him. When he got to Rome he didn't have enough supplies to break down the walls, so he roamed the countryside for 15 years. They raided and burned towns and destroyed crops. Then the Romans attacked Carthage and Hannibal was called home. Then he lost the battle and the war. The Romans got tons of resources, wealth, and new territories including Spain from this.
In the third Punic War the Romans attacked the Carthaginians because they showed signs of regaining power. The Romans burned Carthage and plowed salt into their fields so nothing would grow. The Romans either killed the Carthaginians or sold them into slavery. This ruined Romes reputation of being mild rulers and would lead to their downfall. Rome added Greece to the areas under its rule and became the leading power of the Mediterranean world