In this talk we will report some recent joint work with Jungang Li on sharp critical and subcritical Moser-Trudinger inequalities on complete noncompact Riemannian manifolds under some curvature conditions.Our philosophy is to use an approach from sharp local Moser-Trudinger inequalities on compact domains to deduce sharp global inequalities on complete noncompact Riemannian manifolds using the level sets of functions under consideration and thus avoid the argument of symmetrization which is not available on Riemannian manifolds.