When does a closure implement Fn, FnMut and FnOnce?

The traits each represent more and more restrictive properties about closures/functions, indicated by the signatures of their call_... method, and particularly the type of self:

• FnOnce (self) are functions that can be called once
• FnMut (&mut self) are functions that can be called if they have &mut access to their environment
• Fn (&self) are functions that can be called if they only have & access to their environment

A closure |...| ... will automatically implement as many of those as it can.

• All closures implement FnOnce: a closure that can't be called once doesn't deserve the name. Note that if a closure only implements FnOnce, it can be called only once.
• Closures that don't move out of their captures implement FnMut, allowing them to be called more than once (if there is unaliased access to the function object).
• Closures that don't need unique/mutable access to their captures implement Fn, allowing them to be called essentially everywhere.

These restrictions follow directly from the type of self and the "desugaring" of closures into structs; described in my blog post Finding Closure in Rust.

For information on closures, see Closures: Anonymous Functions that Can Capture Their Environment in The Rust Programming Language.