  Max-stable random fields are very appropriate for the statistical modelling
of spatial extremes. Hence, integrals of functions of max-stable random fields
over a given region can play a key role in the assessment of the risk of
natural disasters, meaning that it is relevant to improve our understanding of
their probabilistic behaviour. For this purpose, in this paper, we propose a
general central limit theorem for functions of stationary max-stable random
fields on $\mathbb{R}^d$. Then, we show that appropriate functions of the
Brown-Resnick random field with a power variogram and of the Smith random field
satisfy the central limit theorem. Another strong motivation for our work lies
in the fact that central limit theorems for random fields on $\mathbb{R}^d$
have been barely considered in the literature. As an application, we briefly
show the usefulness of our results in a risk assessment context.
