A combinatorial Hopf algebra is a graded connected Hopf algebra with finite-dimensional pieces equipped with a character. We will show that the quasi-symmetric (symmetric) functions can be made into a (cocommutative) combinatorial Hopf algebra and that it is the terminal object in the category of (cocommutative) combinatorial Hopf algebras. We will give examples of combinatorial Hopf algebras naturally arising from combinatorics and an application of this result to the study of affine Stanley symmetric functions.