  This paper brings a contribution to the Bayesian theory of nonparametric and
semiparametric estimation. We are interested in the asymptotic normality of the
posterior distribution in Gaussian linear regression models when the number of
regressors increases with the sample size. Two kinds of Bernstein-von Mises
Theorems are obtained in this framework: nonparametric theorems for the
parameter itself, and semiparametric theorems for functionals of the parameter.
We apply them to the Gaussian sequence model and to the regression of functions
in Sobolev and $C^{\alpha}$ classes, in which we get the minimax convergence
rates. Adaptivity is reached for the Bayesian estimators of functionals in our
applications.
