Nice summary of events by /u/cleopatra_philopater. If I could I'm going to take this rare opportunity to add a footnote, and plug my namesake!

Livy (9.16-19) provided a really interesting aside where he examined a "what-if" scenario in which Alexander the Great goes west instead of east and takes on Rome. Of course, as Alexander the Great's ambition was always to go east and keep going, it was never possible he would have gone west instead of east. Perhaps he could have gone west later if he had survived, but he didn't so this remains a total fantasy, but nonetheless a fantasy that occurred not only to military history fanboys/girls on the internet, but also to the ancients.

The section of Livy I referenced above begins with lavish praise of Papirus Cursor, a renowned general who was active at the same time as Alexander the Great, known for his size and strength, his speed when force-marching on foot (hence his cognomen, Cursor), his courage, and his ability and sternness as a general. Livy then proceeds to compare him to Alexander the Great, and diverges into a general comparison between Alexander the Great and his army, and the contemporary Roman generals and their legions.

He sums up the key considerations as "the number and bravery of the soldiers, the abilities of the commanders, and fortune..." Livy makes the fair point that Alexander died young, and could have suffered any number of reversals if he had campaigned longer, as did many greats who lived longer (Pompeius Magnus, Pyrrhus and Hannibal are good examples. Livy lists the prominent Roman generals who were active at the time, of which there are 11 (one of whom was Papirus Cursor), and lapses into a chauvinist anti-eastern diatribe about Roman discipline and skill at war compared to eastern weakness.

Livy then continues on the theme of Alexander's character flaws, his fortune that might have easily reversed over time, mentions there are numerous Roman generals who also never lost battles, and makes the point that while the Macedonians only had one Alexander, the Romans had many generals equal to him.

Livy then compares the armies. In numbers, he compares the 30,000 foot and 4,000 horse of Alexander to the quarter of a million qualified military age men that the Romans possessed at the time, enough to reliably field ten legions at any time. He repeats Polybius' claims about the inflexibility and inferiority of the phalanx to the maniple and mentions the many times Romans faced the Macedonian/Hellenistic phalanx and triumphed.

I can't definitively comment on the accuracy of Livy's analysis, of course war is not so simple as comparing personalities of generals and numbers of soldiers. Livy's analysis is a product of Roman assumptions and arrogance about discipline, weakness and immorality of Greeks, and the even worse weakness and immorality of the Persians. He has the numbers right, but of course Darius had great numbers and it is not quite right to claim that the Persians couldn't fight. Livy probably doesn't quite take into account that the Macedonian army of Alexander was a very different beast to the Hellenistic armies of his successors. The Romans developed their tactics and organisation profoundly in response to battles with Samnites and Gauls in the 300's and the Epirotes and Carthaginians in the 200's. Livy is assuming the effectiveness of the Roman army after the Second Punic War, when it had well and truly learned its business, is comparable to that of the Roman Army in the mid 300's, which was a less flexible force. But that said, Livy is right that the Romans certainly did know how to fight during this time and had superior numbers, so perhaps they could have done it after all.

But mainly Paprius Cursor was really cool.

Source: Read it all here if you like: http://oll.libertyfund.org/titles/livy-history-of-rome-vol-2