  Linear processes on functional spaces were born about fifteen years ago. And
this original topic went through the same fast development as the other areas
of functional data modeling such as PCA or regression. They aim at generalizing
to random curves the classical ARMA models widely known in time series
analysis. They offer a wide spectrum of models suited to the statistical
inference on continuous time stochastic processes within the paradigm of
functional data. Essentially designed to improve the quality and the range of
prediction, they give birth to challenging theoretical and applied problems. We
propose here a state of the art which emphasizes recent advances and we present
some promising perspectives based on our experience in this area.
