This book offers a predominantly theoretical coverage of statistical prediction, with some potential applications discussed, when data and/ or parameters belong to a large or infinite dimensional space. It develops the theory of statistical prediction, non-parametric estimation by adaptive projection with applications to tests of fit and prediction, and theory of linear processes in function spaces with applications to prediction of continuous time processes. This work is in the Wiley-Dunod Series co-published between Dunod (www.dunod.com) and John Wiley and Sons, Ltd.