Given a sequence of i.i.d. random variables with values in a measure space \((T,{\mathcal B})\) and a class F of functions on T which have the Vapnik-Červonenkis property, sufficient conditions are obtained for a normalized empirical process \(\nu_ n\) viewed as a stochastic process indexed by F: \[ \nu(f) = n^{-}\sum_{i\leq n}f(X_ i)-Ef(X_ i),\quad f\in F, \] to converge weakly to the Gaussian process indexed by F. Under mild additional assumptions these conditions are necessary as well. This improves D. Pollard’s theorem [J. Aust. Math. Soc., Ser. A 33, 235-248 (1982; Zbl 0504.60023)] and presents necessary and sufficient conditions for the central limit theorem for weighted empirical processes indexed by Vapnik-Červonenkis classes of sets.