While during most of the late Republic and early Empire, a veteran legion seems to have been practically invincible in battle. However, there are a few very prominent defeats that occurred during this period, such as Gergovia, Suthul, Carrhae, Halys, or Mount Garganus. While in the narrative histories this was blamed on the generals, what credit was due to the enemies of the Romans in these defeats? What military advances did they implement to overcome Rome's superior military?
The Battle of the Teutoburg Forest is generally regarded by modern historians as "Rome's Greatest Defeat". The leader of the Anti-Roman alliance was Arminius (aka Hermann the German - yes that was a nickname given to him later on in History). Arminius, though Germanic born, was a Roman citizen and had been trained by the Roman Military. So he had a fairly thorough understanding of their tactics. Due to this defeat, the Romans ceased expansion beyond the Rhine and Danube Rivers. Arminius had superior numbers and knew where to allocate them because of his understanding of Roman tactics.
The Macedonian King Phyrrus' invasion of the Southern Italian peninsula made use of war elephants to dramatic effect in the battle of Heraclea in 280 BCE, Routing Roman forces. It one of the first experiences Romans had with the beasts and was very successful initially, particularly in their use against cavalry as horses are naturally terrified of elephants. The Romans quickly developed counter-measures (the famous flaming pigs) that rendered Phyrrus' elephants far less effective.
Quick Fact: it is from this conflict that the term that we get 'Phyrric victory'. Following one of Phyrrus' early victories against the Romans in which sustained devastating losses he is anecdotally quoted as having said "If I win another battle against these Romans, I will be ruined"