The principle of the curved sword is that you have a longer blade in a smaller space - so it has a smaller swinging arc with the same length of blade. There are advantages and disadvantages to a smaller swinging arc - some of the advantages are that you can get inside the arc of an opponent with a larger arc, or you can fight in more confined space (allowing more closely packed fighting units). It has disadvantages too - smaller arc means shorter reach, for instance.
The other feature of the curved blade is that the curve imparts more effect to a slicing action, and it is more suitable for slicing or slashing than a straight blade, which is more suitable for hacking or chopping actions (and stabbing, in the case of swords). Slicing is faster, more accurate, and takes less exertion than hacking. The reason Europeans didn't start using curved blades until relatively late in their history was that you can't slice armour, it must be hacked or pierced by a stabbing action.
Besides the disappearance of armour, European light cavalry frequently adopted curved sabres because they were better at slicing and slashing, which you can do from horseback while moving at speed (chopping and stabbing with blades is hard to do at anything faster than a trot). Heavy cavalry - which fought in more tightly packed formations - retained the straight blade, because in a close formation, cavalry had to use a vertical swinging arc and metal helmets cannot be sliced.
I've been training in traditional sword combat (not fencing, actual swords) and my mentor recently introduced me to Arabian-styled swords with curved hilts. They're very interesting, but I'm not sure what the advantages are. Help?
