This is intended to complement the recent talk of Pham Huu Tiep in this seminar but will not assume familiarity with that talk. The estimates in the title are upper bounds of the form |χ(g)|≤χ(1)α, where χ is irreducible and α depends on the size of the centralizer of g. I will briefly discuss geometric applications of such bounds, explain how probability theory can be used to reduce to the case of elements g of small centralizer, discuss the level theory of characters, and conclude with the reduction to the case of characters χ of large degree. For such pairs (g,χ), exponential character bounds are trivial.