Modern technology has facilitated the collection of data whose observations are functions rather than numbers or vectors. Functional data are increasingly observed in many fields; these include meteorology, auxology, biology, chemometrics, among others. The analysis of this type of data is known as functional data analysis (FDA).

FDA is still in its initial stages and lots of work has to be done to develop the classical statistical tools to deal with functional data. Our aim in this dissertation is to discuss statistical analysis, especially regression, when data are available in functional form.

In the full-functional simple regression model, where the response and covariate are functions, estimation of the regression surface is an ill-posed problem due to the infinite-dimensionality of the covariate function which produces infinitely many solutions. Hence, we describe a certain generalization of the full-functional simple regression model, namely; the functional reduced-rank regression (FRRR). We will show how FRRR can be used to reduce dimensionality and improve estimation in the full-functional regression model. We will also show that FRRR provide an alternative approach to functional principal components analysis (FPCA), where the problem of estimating the regression surface with reduced-rank is related to the problem of deciding how many principal components to use in any function. We will show that functional canonical correlation analysis (FCCA) can also be regarded as a special case of the FRRR, where the problem of estimating the regression surface with reduced-rank is related to the problem of deciding how many pairs of canonical variates to use with any pair of functions.

The rest of this thesis is devoted to the extension of FRRR to deal with vectors of functions. We begin by introducing the functional multiple and multivariate regression models. Methodologies are proposed for several connected topics, including multivariate FPCA and multivariate FCCA. The methods developed are applied to the Dow Jones Industrial data, in addition to well studied data sets in the FDA literature; including Canadian temperature data and gait data.