Mature, grown-up, old enough (with the implication that one can look after oneself and make one's own decisions).
Of a person: having obtained worldly experience, and lacking naiveté; cosmopolitan, worldly-wise.
Complicated, especially of complex technology.
Of art or other things: appealing to the tastes of an intellectual or sophisticated (sense 1) person; cerebral; also, cultured, elegant, refined.