There are 13 Royal Arms Locations in Final Fantasy XV. Finding all 13 unlocks the “Faithful Heir” trophy / achievement.
How to find the Royal Arms in Final Fantasy XV
Each Royal Arm in Final Fantasy XV has unique attack patterns and stat boosts. They can be used in combat by Noctis. They are more powerful than normal weapons but using them depletes your own health. Even if you don’t want to use these weapons you should equip them for the stat boosts they provide. Furthermore, the Royal Arms bypass all enemy resistances and always deal full damage (no enemies are resistant to them).
The first 7 Royal Arms are automatic / unmissable story unlocks. You get them automatically during main quests. The remaining 6 Royal Arms are from dungeons and Royal Tombs which you must find on your own. None of them are missable. There is free roam after finishing the story and you can go back everywhere.
The dungeons are fairly difficult, one of them being for level 55+. If you want you can do them before leaving for chapter 9 but it will be very difficult. We highly recommend you finish the story first to get stronger. Remember, you can switch to easy difficulty in the game options at any time (in easy mode you cannot die).
To find the remaining 6, just remember to beat these dungeons. Bow of the Clever is obtained in Balouve Mines. Sword of the Tall is obtained in Costlemark Tower. Shield of the Just is obtained in Tomb of the Just. Scepter of the Pious is obtained in Malmalam Thicket. Mace of the Fierce is obtained in the Rock of the Ravatogh. Star of the Rogue is obtained in The Myrlwood.