Subjectively... When they feel they can live alone. By this, I mean that when someone thinks that if their parents disowned them, they would have the skills necessary to survive, then they should be considered adults.
In more objective terms (and more clarifying declarations), a person should be considered an adult after passing a set number of critical objectives. This would include:
1) Being able to successfully find a job and adequately balance a job with other aspects of their life.
2) Being able to think abstractly/logically/etc.
(Piaget's fourth stage of development).3) Realizing and being able to interact with others to live a full life.
4) Realizing the importance that present decisions play in the future.
I guess my objective suggestion wasn't that much better than the subjective one, but it's so hard to put an absolute age on something like this. I think I matured (in some ways) enough to have been an "adult" at 16. I was a supervisor at the place of my employment, I was responsible enough to balance school and work, and I could interact with others in a mature and adult fashion. But I've seen 20 year olds who are less mature and less worthy of "adulthood" than especially mature 14 year olds... If I was forced to put an age on it, I'd go ahead and say adulthood at 17 here in America. Even America is too diverse for this to be an easy thing to define, and trying to apply a universal age of adulthood across different cultures would be ridiculously myopic. And since 17 would mean adulthood, it should include all benefits and shortcomings of being an adult (e.g. drinking alcohol, fulfilling federal and state obligations, etc.).