Final answer:
The Carthaginians, led by Hannibal, won important battles in Italy, including at Trebia, Lake Trasimene, and Cannae, but were ultimately defeated by Rome at the Battle of Zama in 202 BCE.
Step-by-step explanation:
Carthaginian soldiers did win multiple battles on Italian soil during the Second Punic War, notably under the leadership of the general Hannibal Barca. The most significant victories were at the Battle of Trebia (218 BCE), the Battle of Lake Trasimene (217 BCE), and the Battle of Cannae (216 BCE), where Hannibal's army inflicted heavy losses on the Romans. Despite these victories, Carthage was unable to secure a decisive defeat over Rome, and, ultimately, the Roman general Scipio Africanus defeated Hannibal at the Battle of Zama in 202 BCE, which led to the end of the Second Punic War with Roman victory.
There was also the Third Punic War, which was one-sided and resulted in the complete destruction of Carthage in 146 BCE. During their encounters, Rome and Carthage fought for dominance over the Mediterranean, starting with the First Punic War that began in 264 BCE and ended with Carthage losing Sicily and paying war indemnity to Rome.