Extreme value methods are used in a wide range of applications, for example they may be used for modelling wave heights and river levels in hydrology, wind speeds in structural engineering and share price return levels in economics. Many statistical models and methods of inference exist for the extreme values of univariate sequences of independent and identically distributed (IID) random variables. However, in most applications, the data sets are not IID and are often multivariate, and yet methods for modelling the extremes of sequences which fail to fulfil one, or both, of the IID assumptions and (or) are multivariate remain the subject of ongoing research. The work contained in this thesis is a contribution to this area. Most of our work has been motivated by a multivariate air pollution data set, which shows complex seasonal trends and covariate relationships. We begin with a model for the extremes of a univariate sequence which displays short-range dependence within the sample extremes. Next we propose a method for modelling the extremes of a non-stationary univariate process; we then extend this methodology to model a multivariate process with non-stationary marginal and dependence structures. Finally we consider a new estimator for the dependence structure of a sequence of multivariate extremes which are pairwise dependent.