1. Battle of the Trebbia
The Battle of the Trebbia was fought in Italy, in 218 B.C., between forces led by Sempronius Longus and Hannibal. Sempronius Longus' 36,000 infantry were arrayed in a triple line, with 4000 cavalry on the side; Hannibal had a mixture of African, Celtic, and Spanish infantry, 10,000 cavalry, and his notorious war elephants in front. Hannibal's cavalry broke through the lesser numbers of the Romans' and then attacked the bulk of the Romans from the front and sides. Hannibal's brother's men then came up from hiding behind the Roman troops and attacked from behind, leading to the defeat of the Romans.
Source: John Lazenby "Trebbia, battle of" The Oxford Companion to Military History. Ed. Richard Holmes. Oxford University Press, 2001.
Following the loss, the Romans appointed Fabius Maximus dictator. Fabius Maximus was called the delayer, cunctator because of his perceptive, but unpopular policy of refusing to be drawn into pitched battle.
Reference: John Lazenby "Lake Trasimene, battle of" The Oxford Companion to Military History. Ed. Richard Holmes. Oxford University Press, 2001.
In 216 B.C., Hannibal won his greatest victory in the Punic War at Cannae on the banks of the Aufidus River. The Roman forces were led by consul Lucius Aemilius Paullus. With a substantially smaller force, Hannibal encircled the Roman troops and used his cavalry to crush the Roman infantry. He hamstrung those who fled so he could later return to finish the job.
Livy says 45,500 infantry and 2700 cavalry died, 3000 infantry and 1500 cavalry taken prisoner.
"Of the infantry ten thousand were taken prisoners in fair fight, but were not actually engaged in the battle: of those who were actually engaged only about three thousand perhaps escaped to the towns of the surrounding district; all the rest died nobly, to the number of seventy thousand, the Carthaginians being on this occasion, as on previous ones, mainly indebted for their victory to their superiority in cavalry: a lesson to posterity that in actual war it is better to have half the number of infantry, and the superiority in cavalry, than to engage your enemy with an equality in both. On the side of Hannibal there fell four thousand Celts, fifteen hundred Iberians and Libyans, and about two hundred horse."
The Battle of Zama or simply Zama is the name of the final battle of the Punic War, the occasion of Hannibal's downfall, but many years before his death. It was because of Zama that Scipio got to add the label Africanus to his name. The exact location of this battle in 202 B.C. is not known. Taking lessons taught by Hannibal, Scipio had substantial cavalry and the help of former allies of Hannibal. Although his infantry force was smaller than Hannibal's, he had enough to get rid of the threat from Hannibal's cavalry -- with the fortuitous help of Hannibal's own elephants -- and then circle around to the back -- a technique Hannibal had used in earlier battles -- and attack Hannibal's men from the rear.
Source: John Lazenby "Zama, battle of" The Oxford Companion to Military History. Ed. Richard Holmes. Oxford University Press, 2001.