Battle of the Granicus
The modern observations that the river is not difficult to cross contradicts the ancient historians, who emphasized the hardship of the fording of the river. The ancient historians might have exaggerated to make Alexander's accomplishment seem more formidable.

Opposing forces: 

Macedonian army: The Macedonian army consisted mostly of infantry. The heavy infantry numbered 12,000 and included the Foot Companions and a smaller group of elite hypaspists. The light infantry totalled 1,000 and contained archers and elite Agrianian javelin-men. The heavy cavalry was made up of 1,800 elite Companion cavalry, 1,800 Thessalian cavalry and 600 Greek allied cavalry. The light cavalry numbered 900 in total and was composed of prodromoi, Paeonians and Thracians. Alexander also possessed Greek allied and mercenary infantry, but he had marched to the Granicus without them. In total, Alexander's army numbered 12,000 heavy infantry, 1,000 light infantry and 5,100 cavalry for a total of 18,100 men.
From the right wing to the left, Alexander placed seven of his eight squadrons of Companion cavalry, along with the archers and Agrianians, under the command of Philotas. Because the Agrianians usually fought interspersed with the Companion cavalry, they were most likely placed in front of them. The prodromoi, the Paeonian cavalry and the eighth squadron of the Companion cavalry under Socrates were posted next to them. Overall command of these three cavalry units was held by Amyntas, son of Arrhabaeus. After them came the hypaspists commanded by Nicanor, which guarded the right flank of the Foot Companions next to them. There were six units of Foot Companions: led by Perdiccas, Coenus, Craterus, Amyntas the son of Andromenes, Philip the son of Amyntas, and Meleager. To the left of Meleager's Foot Companions were the Thracian cavalry under the command of Agathon. Next to them were stationed the allied Greek cavalry under Philip, son of Menelaus. At the extreme left were the Thessalian cavalry under the command of Calas. Alexander placed Parmenion in overall command of the Foot Companions of Philip and Meleager, and the cavalry on the left wing. Alexander commanded the remainder of the army on the right wing. The Greek allied and mercenary infantry, which were part of the invasion force, were not used during the battle.

Persian army: For the Persian army Arrian gives a number of almost 20,000 infantry, all Greek mercenaries, and 20,000 cavalry. Diodorus gives a figure of 10,000 cavalry and 100,000 infantry. Justin gives an even higher number of 600,000 men in total for the Persians. The historian Azar Gat explains that although the Persian Empire and its armies were large, the size of its armies were wildly exaggerated in the Greek sources. Gat adds that this was an "invariable" habit of pre-modern writers, as they both lacked precise information and were patriotically biased.
N. G. L. Hammond accepts the Persian troop numbers given by Arrian. A. M. Devine and Krzysztof Nawotka regard the numbers given by Justin and the number for infantry given by Diodorus to be fantastical because such a large army would have been a logistical impossibility. Both historians consider the figure of 10,000 cavalry given by Diodorus to be more plausible. J. F. C. Fuller also considers a number of 10,000 cavalry to be more likely, but he does not mention the figure of Diodorus. Arrian's claim on the strength and composition of the Persian infantry has elicited more detailed discussion. There have been arguments that the figure of 20,000 is accurate, but also included non-Greek native infantry. Others have argued that Greek mercenaries may have been the only infantry present but were not as numerous.
Krzysztof Nawotka considers Arrian's claim of 20,000 infantry credible, but does not think it consisted entirely of Greek mercenaries. Such a large force of mercenaries would have been raised only very occasionally, after a long period of preparation and under the command of the Great King. Because the Achaemenids had not made such preparations for the battle, he judges that only a small fraction of the infantry were Greek mercenaries. Accordingly, local detachments of little military value made up the vast majority of the infantry. Based on Plutarch's statement that the infantry of both sides engaged after the Macedonian phalanx crossed the river, Ernst Badian likewise believes that non-Greek infantry was part of the Persian army and posted directly behind the cavalry. In this light, he considers 20,000 a reasonable estimate for the total number of infantry. He believes that Arrian's claim that all Persian infantry consisted of Greek mercenaries was either a mistake or an attempt to enhance Alexander's glory.
A.M. Devine argues that Arrian likely exaggerated the number of Greek mercenaries for propaganda purposes. He notes that the only other information on Greek mercenaries in Asia Minor at that time is given by Diodorus and Polyaenus. Diodorus writes that Memnon's force of Greek mercenaries, which Darius III had sent to Asia Minor to fight the Macedonian expeditionary force, numbered 5,000 men. Polyaenus gives a lower number for these, 4,000. Assuming that Arrian was right in his claim that the infantry consisted exclusively of Greek mercenaries and that the figure of 10,000 cavalry given by Diodorus is plausible, he suggests that the Persian army likely numbered 14,000 to 15,000 men. This was significantly inferior both in numbers and quality compared to the 18,100 men fielded by Alexander. Like A. M. Devine, J. F. C. Fuller thinks the number of 5,000 mercenaries given by Diodorus is more realistic, but he thinks there must have also been a considerable force of local levies.
A.M. Devine goes on to argue that this inferiority of the Persian force would explain why the Persians chose to defend a river crossing and place their cavalry in the front line. It would also explain why Alexander decided to risk a frontal attack there. Likewise, the encirclement and massacre of the Greek mercenaries later during the battle would have been unlikely if they had actually numbered 20,000 instead of 4,000 or 5,000. Finally, the low estimate for the size of the Persian army would fit with the small number of casualties for the Macedonians.
The Persians stationed their cavalry along the bank of the Granicus. The left wing was held by Memnon and Arsames, each with their own cavalry units. Next to them Arsites commanded the Paphlagonian horsemen. Spithridates was placed next to him and led the Hyrcanian cavalry. The center was held by cavalry of unspecified ethnicity. The right wing consisted of 1,000 Median cavalry, 2,000 cavalry headed by Rheomithres and another unit of 2,000 Bactrian horse. The Persians moved some of their cavalry to reinforce their left wing when they noticed Alexander himself was stationed opposite this wing. The Greek mercenaries were placed behind the cavalry and led by the Persian Omares.
Historians have criticized several decisions made by the Persian commanders. One of these is the deployment of the cavalry on the river's bank, because it did not allow the cavalry to charge. This has been explained by the fact that the Persian cavalry at that time did not use charge tactics yet. Instead, they would approach enemy formations and attack them with javelins to disrupt them from a distance. Once the formation was sufficiently disordered, the cavalry would switch to curved swords for the melee fighting. Likewise, historians have questioned why Memnon was given command of a cavalry unit at the front line, given his success with leading the Greek mercenaries. A possible motive is that the Persian commanders were jealous of him and did not trust him. Memnon would have been easier for them to control in their midst at the front line.  A third point of criticism is the deployment of their infantry far from the frontline. Modern historians have suggested that the Persian commanders did so out of resentment towards the Greek mercenaries, which had defeated the Macedonians at Magnesia the year prior. By defeating Alexander with just their cavalry, they would have regained their pride.

Battle: There are three different accounts of the battle given by the ancient historians Arrian, Plutarch and Diodorus Siculus. Arrian's account is the longest and most detailed, Plutarch's account is shorter but corresponds with Arrian's narrative except for some small differences. The account of Diodorus Siculus however is fundamentally different from that of Arrian and Plutarch.

Account of Arrian: After both armies finished their deployment, there was a moment of silence. Alexander then ordered Amyntas, son of Arrhabeus, to attack the Persian left wing with the Companion cavalry squadron of Socrates (now led by Ptolemy, son of Philip) at the front, followed by the Paeonian cavalry, the prodromoi and an unspecified infantry unit.