Short answers, pulled from the story.
The campaign history of the Roman military spans from the 8th century BC, when Rome was a small city-state on the Italian Peninsula, to the fall of the Western Roman Empire in the 5th century AD. It encompasses land battles fought from Britain and Gaul in the west to Armenia and Mesopotamia in the east, including the Punic Wars, the Samnite Wars, the Pyrrhic War, wars against Macedon, the Gallic Wars, and a long sequence of civil wars.
Before the First Punic War in 264 BC, Rome had no real navy, as all prior wars had been fought on land in Italy. Forced by the conflict with Carthage to build a fleet from scratch, Rome suffered early disasters such as the Battle of the Lipari Islands. Within four years, after inventing a grappling device called the Corvus, a Roman fleet under C. Duillius defeated Carthage at the Battle of Mylae, and Rome went on to win the First Punic War through naval victory at the Battle of the Aegates Islands.
Hannibal was a member of the Barcid family of Carthaginian nobility who launched the Second Punic War by attacking Saguntum, a city with diplomatic ties to Rome. He raised an army in Iberia, crossed the Italian Alps with elephants, and invaded Italy, winning major victories at Ticinus in 218 BC, the Trebia, Lake Trasimene, and Cannae. Despite inflicting some of the worst defeats Rome ever suffered, he was ultimately recalled to Africa and defeated by Scipio Africanus Major at the Battle of Zama.
The Battle of the Teutoburg Forest in 9 AD was a catastrophic Roman defeat in which the Germanic leader Arminius destroyed the forces of Varus almost to the last man. Despite the loss, Rome recovered and sent Germanicus on further campaigns against the Germanic tribes, though the Rhine eventually became the permanent limit of Roman expansion to the north.
The Social War broke out in 91 BC because Rome's Italian allies, known as the Socii, shared the military burdens of Rome's campaigns but received none of the rewards, including Roman citizenship. Although Roman troops won decisive engagements such as the Battle of Asculum, the Socii achieved their core objective through law: the Lex Julia and Lex Plautia Papiria granted citizenship to more than 500,000 Italians.
The Pyrrhic War ran from 280 to 275 BC, when Pyrrhus of Epirus landed a Greek army of some 25,000 men and war elephants in Italy at the request of the Greek colony of Tarentum. Pyrrhus won at Heraclea and Ausculum but found Rome unwilling to negotiate while his army remained on Italian soil. Facing unsustainable losses, an inability to find new Italian allies, and a Roman alliance with Carthage, Pyrrhus withdrew. At the Battle of Beneventum in 275 BC, the Romans used javelins, fire, and direct blows to neutralise the elephants, and Pyrrhus abandoned the peninsula entirely.