There are some items which require your character to use both hands. These are only certain ones though.
You cannot dual wield with any weapon. Only specific ones.
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Yes, but only certain weapons can. These weapons are labeled as 'Akimbo' when they are being dual-wield.
In order to wield dragon weapons, you need to have high enough levels (usually level 60) and in most cases a quest. To wield different weapons, you need to do different quests. For instance: to wield dragon long-swords or daggers, you need to do the lost city quest. but in order to wield the dragon scimitar, you need to do the monkey madness quest. The runewiki has a list of the quests required to wield the different dragon weapons.
To dual wield, put a weapon in your main hand and in your off hand too. The main hand weapon can be any kind of weapon as long as it is a single hand weapon - so no two-handed axes, hammers, or magic staves, or bows. The offhand weapon must be a dagger. However if you unlock the Dual-Weapon Mastery talent, then any single hand weapon can be put in the offhand slot.
If you have a weapon in your inventory, click on it to wield it. This will un-wield any other weapon you have in your weapon slot.
After you get your first duel spirit, you get to duel him. There are also some other times after that where you duel him