  We develop the concept of character level for the complex irreducible
characters of finite, general or special, linear and unitary groups. We give
characterizations of the level of a character in terms of its Lusztig's label
and in terms of its degree. Then we prove explicit upper bounds for character
values at elements with not-too-large centralizers and derive upper bounds on
the covering number and mixing time of random walks corresponding to these
conjugacy classes. We also characterize the level of the character in terms of
certain dual pairs and prove explicit exponential character bounds for the
character values, provided that the level is not too large.
