Extreme values are by definition rare, and therefore a spatial analysis of extremes is attractive because a spatial analysis makes full use of the data by pooling information across nearby locations. In many cases, there are several dependent processes with similar spatial patterns. In this paper, we propose the first multivariate spatial models to simultaneously analyze several processes. Using a multivariate model, we are able to estimate joint exceedance probabilities for several processes, improve spatial interpolation by exploiting dependence between processes, and improve estimation of extreme quantiles by borrowing strength across processes. We propose models for separable and non-separable, and spatially continuous and discontinuous processes. The method is applied to French temperature data, where we find an increase in the extreme temperatures over time for much of the country.