If you want to know how weapon mastery works, it more or less serves as the weapon system in New World that encompasses everything from weapon skills and what weapons you can use.
For starters, if you want to use a specific weapon in the game, you will need to level the weapon mastery for that weapon. That being said, there appear to be around nine weapon masteries so far that include ones for axes, swords, guns, spears, and even war hammers and bows.
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So if any of these take your fancy, you will need to select it, and by selecting, we mean simply equipping it and start using it. Once a weapon is equipped, you can level the weapon mastery it belongs to by killing monsters and other enemies. Every time you kill something, you will gain XP, so the more XP you earn, the faster you will level up your masteries.
Each of the masteries will also have two skill trees, and the more you level them up, the more abilities you can unlock.
You can unlock passives and other bonuses by leveling the masteries up and then using a point. Also, every time you level up, you will gain one additional point that can be spent towards new perks, abilities, and whatnot.