Let P{\mathcal{P}} be a nonparametric probability model consisting of smooth probability densities and let [^(p)]_n{\hat{p}_{n}} be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law \mathbbP{\mathbb{P}} . With [^(\mathbbP)]_n\hat{\mathbb{P}}_{n} denoting the measure induced by the density [^(p)]_n{\hat{p}_{n}} , define the stochastic process [^(n)]_n: f® Ön òfd([^(\mathbbP)]_n -\mathbbP){\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P}) where f ranges over some function class F{\mathcal{F}} . We give a general condition for Donsker classes F{\mathcal{F}} implying that the stochastic process [^(n)]_n\hat{\nu}_{n} is asymptotically equivalent to the empirical process in the space l^¥(F){\ell ^{\infty }(\mathcal{F})} of bounded functions on F{ \mathcal{F}} . This implies in particular that [^(n)]_n\hat{\nu}_{n} converges in law in l^¥(F){\ell ^{\infty }(\mathcal{F})} to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes F{\mathcal{ F}} . We give a number of applications: convergence of the probability measure [^(\mathbbP)]_n{\hat{\mathbb{P}}_{n}} to \mathbbP{\mathbb{P}} at rate Ön{\sqrt{n}} in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; Ön{\sqrt{n}} -efficient estimation of nonlinear functionals defined on P{\mathcal{P}} ; limit theorems at rate Ön{\sqrt{n}} for the maximum likelihood estimator of the convolution product \mathbbP*P{\mathbb{P\ast P}} .