  Max-stable processes are natural models for spatial extremes because they
provide suitable asymptotic approximations to the distribution of maxima of
random fields. In the recent past, several parametric families of stationary
max-stable models have been developed, and fitted to various types of data.
However, a recurrent problem is the modeling of non-stationarity. In this
paper, we develop non-stationary max-stable dependence structures in which
covariates can be easily incorporated. Inference is performed using pairwise
likelihoods, and its performance is assessed by an extensive simulation study
based on a non-stationary locally isotropic extremal $t$ model. Evidence that
unknown parameters are well estimated is provided, and estimation of spatial
return level curves is discussed. The methodology is demonstrated with
temperature maxima recorded over a complex topography. Models are shown to
satisfactorily capture extremal dependence.
