I will survey some applications of Donaldson's technique of quantitative transversality of "approximately holomorphic" functions in symplectic geometry.  I will explain the basic terms and present the main ideas of the technique.  Donaldson used it to show that the Poincare dual of any sufficiently large multiple of an integral symplectic form is represented by a symplectic submanifold.  Another application is joint work with E. Giroux in which we prove the existence of Lefschetz fibrations on certain symplectic manifolds.