So obviously this is an incredibly broad question, but I'm thinking specifically around the Mediterranean and the from ~500 to ~1400 CE. I'm no expert on naval warfare during any era, but based on what I know it seems like you don't see many massive naval engagements like the Battles of Salamis and Actium during the Middle Ages. If this is the case, was it a result of naval combat becoming less central to warfare as a whole, a dearth of polities that could field large armadas like the Romans and Persians did, or something else entirely?
Additionally, did the methods of naval warfare change dramatically during this period? My understanding of classical naval combat was that it relied heavily on ramming, boarding and using soldiers equipped with bows, javelins, etc to rain missiles on opposing vessels. Was this also the case in the area during the Middle Ages, or did new technologies and tactics dramatically change the way ships engaged one another (I know, for example, that the Arabs had invented effective torpedoes by some point in the 13th century)?
I am not familiar enough with naval warfare in antiquity to tell you how it changed, but this write up by /u/EyeStache in the FAQ gives a good picture of naval warfare among the Vikings during the middle ages.
There are a few reasons why it make seem that there isn't as much naval combat in the Mediterranean after the Classical era, heading into late antiquity and beyond. This isn't really true- William Ledyard Roger's Naval Warfare Under Oars covers several important engagements and uses of seapower in this time period, which we will talk about in a bit.
The first reason that it doesn't seem there's much naval action is that there are fewer sources that attest to naval combat in this era than in the Classical era. Part of this is an issue of source generation- there were fewer things written down, particularly in the earlier part of your time period than in earlier eras. Also, a great deal of what was written down comes from Arabic sources, which use a different language, historical style and approach to what was important than Greek and Roman sources, and, for some time, were eiter inaccessable to or disregared by, western historians.
Second, we have to consider that navies aren't simple things to build and maintain. Ships, and warships in particular, are technologically sophisticated and expensive to build, maintain and use. This means, to raise any navy of any size requires, first, a wide variety of tradesmen- shipwrights, sailmakers, carpenters, blacksmiths, rope makers, and so on- that aren't really needed to build more pressing things like houses or scythes to bring in the harvest, which implies a level of economic sophistication and food security that became increasingly rare, particularly in the Western Mediterranean after the end of the Western Empire. Secondly, you need enough stuff to build ships- wood of the right sorts, cloth, iron, fibers to make ropes, and so on. Thirdly- and this is even more difficult- you need money to pay everyone to gather the materials in one place and then work to build ships, and to keep them at it. Fourth, you need to put together a crew of sailors- a specific trade all on its own- in order to man the ship. In order to have sailors to recruit from, you almost always need seagoing trade nearby, and there is a lot less seagoing trade in this era than in the classical era. Finally, in order to keep the navy going, you need trained and skilled administrators who know how to perform the tasks required to keep ships afloat, to track their movements, and manage the money to build new ships and maintain the old ones.
What this comes down to is that, in this era, only a few polities could really afford much of a navy to begin with. Therefore, we see fairly long stretches were areas of the Mediterranean don't really have a dominant power, or are controlled by significantly wealthy, well organized polities with access to sailors and shipbuilders: The Byzantines, the Arabs, the Venitians, and, a little bit out of your range, but still relevant, the Ottomans. If you're going to have thrilling naval combat, you need two navies that are willing to fight each other- usually, this means they need to be fairly well matched in order to risk ships in combat. So, you get a series of naval actions in the 600s, such as the Battle of the Masts, a lot of fighting around Greece, Crete and Cyprus, and a couple sieges of Constantinople. You also get some naval fighting amongst Italian city-states in the 1000s and 1100s, and the series of battles fought in the War of the Sicilian Vespers by Roger di Lauria. Many of these battles tend to be on the small side- dozens, not hundreds of ships. However, we rarely have anything like good numbers for even the most significant battles, like the Battle of the Masts, which probably involved hundreds of ships.
During this era, and perhaps even before, during the Roman Empire, the ram began to fall out of favor as a weapon of naval combat. Most combat at sea before the 670s involved bringing ships together in line abreast- or some variation, such as a cresent or weighted line- and driving into each other for missile, and then boarding, combat. There are several examples of ships lashing themselves together in order to preven the enemy from interprenetrating the fighting line, which of course rather removes the possibility of manuever. However, in the 670s, the Byzantines introduced what we call Greek Fire- an incendiary formula that burned even when wet. This allowed the Byzantines to sail near enemy ships, and pump a mixture of sea water and Greek fire as a kind of flame thrower from the bow of their ships. Greek Fire, of course, could be used from land installations and during sieges as well, but that's not really important here. Greek Fire wasn't a wonder weapon, though- it required favorable conditions to employ, and the enemies of the Byzantines started wetting their decks before combat, or using wet hides or similar schemes to protect their decks. Still, one did not start a fight with the Byzantines between 600 and 1200 or so without at least thinking about a firey death.
While a little out of range of the dates you listed, it's worth knowing that cannon came to sea in the 1400s and 1500s, which changed Mediterranean warfare a bit. Because the galley remained the warship of choice, it wasn't possible to mount guns broadside the way North Atlantic ships would- the oars had to go along the broadside. So, galleys tended to mount one, or a few, rather heavy guns in the bow. These guns would fire as the ships approached each other, again in line ahead, with hopes of landing a heavy, damaging hit on the enemy before manuevering to board and fight.
If you're looking for some good reading on the subject, I'd suggest looking at William Leadyard Rodger's Greek and Roman Naval Warfare, and his Naval Warfare under Oars, John F. Guiilmartin's Gunpowder and Galleys and Lawrence Mott's Sea Power in the Medieval Mediterranean, which does an execellent job of discussing the administration manning of Roger of Lauria's fleet.
Hope this helps.