If an object becomes a copy of an object that has an Adventure, the copy also has an Adventure. If it changes zones, it will either cease to exist (if it's a token) or cease to be a copy (if it's a nontoken permanent), and so you won't be able to cast it as an Adventure.
Casting a card as an Adventure isn't casting it for an alternative cost. Effects that allow you to cast a spell for an alternative cost or without paying its mana cost may allow you to apply those to the Adventure.
If an effect copies an Adventure spell, that copy is exiled as it resolves. It ceases to exist as a state-based action; it's not possible to cast the copy as a permanent.
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You must still follow any timing restrictions and permissions for the permanent spell you cast from exile. Normally, you'll be able to cast it only during your main phase while the stack is empty.
If an effect instructs you to choose a card name, you may choose the alternative Adventure name. Consider only the alternative characteristics to determine whether that is an appropriate name to choose.
If you cast an adventurer card as an Adventure, use only its alternative characteristics to determine whether it's legal to cast that spell. For example, if Giant Killer is exiled with the last ability of Vivien, Champion of the Wilds, you can't cast it as Chop Down.
If an adventurer card ends up in exile for any other reason than by exiling itself while resolving, it won't give you permission to cast it as a permanent spell.
If a spell is cast as an Adventure, its controller exiles it instead of putting it into its owner's graveyard as it resolves. For as long as it remains exiled, that player may cast it as a permanent spell. If an Adventure spell leaves the stack in any way other than resolving (most likely by being countered or by failing to resolve because its targets have all become illegal), that card won't be exiled and the spell's controller won't be able to cast it as a permanent later.
An adventurer card is a permanent card in every zone except the stack, as well as while on the stack if not cast as an Adventure. Ignore its alternative characteristics in those cases. For example, while it's in your graveyard, Giant Killer is a white creature card whose mana value is 1. It can't be the target of the triggered ability of Mystic Sanctuary.
When casting a spell as an Adventure, use the alternative characteristics and ignore all of the card's normal characteristics. The spell's color, mana cost, mana value, and so on are determined by only those alternative characteristics. If the spell leaves the stack, it immediately resumes using its normal characteristics.
