Starting Chapter 2 Season 5, Quests were added to Battle Royale. These quests are a completely new form of challenges that can reward XP, Bars, or extra styles for cosmetics. Some Quests are automatically given while others can be obtained from Characters in game from interacting with them. There are five different types of Quests which are shown by the rarity/color of the quest:
Legendary Quests are Weekly quests which are timed quests. They are given the week to be completed.
Epic Quests are Weekly or Special quests which are not given a time limit, and can be completed any time during the season.
Rare Quests are Punch Card quests. Once the challenge is completed, the challenge is given again with a more challenging number.