Functional datasets become each day more and more present in many fields of applied sciences (see the contribution by (Delsol et al. 2011 [85]) in this volume) and the statistical community has started to develop specific tools for analyzing them. This active field of modern statistics is known as Functional Data Analysis (FDA). Even if first works in the literature trace back to earlier times, FDA became really popular at the end of the nineties with the book by Ramsay and Silverman (see (Ramsay & Silverman 2005 [267])). This great interest in FDA and the wide scope of its various facets can be seen through recent monographs or handbooks (Ramsay & Silverman 2005 [267], (Ferraty & Vieu 2006 [126]), and (Ferraty & Romain 2011 [124])) and through various special issues in top level statistical journals as Statistica Sinica (2004, vol. 14), Computational Statistics and Data Analysis (2007, vol. 51), Computational Statistics (2007, vol. 22), or Journal of Multivariate Analysis, (2010, vol. 101). The aim of this contribution is to present the state of art in statistical methodology for FDA. Given the increasing interest in the literature in this topic, this bibliographical discussion cannot be exhaustive. So, it will be concentrated on the precursor works and on the most recent advances. Section 17.2 gives special attention to the various

spectral techniques that are widely used as preliminary benchmark methods in FDA. Then Section 17.3 presents the recent developments for exploratory analysis of functional data, while Section 17.4 discusses contributions in problems where functional data are used as explanatory variables. Section 17.5 discusses miscellaneous issues and Section 17.6 sums up the major issues remaining to be considered.