This research will contribute to the analysis of modern and complex data observed as functions or curves. One example of such data is provided by yield curves used in economics as an indicator for future growth, inflation and interest rate expectations, and investor sentiment. Since data of this type is often observed across time, there will be a focus on developing theoretically justified and empirically validated forecasting algorithms, which can find applications in many fields of inquiry including finance, where practitioners are interested in predicting the volatility curves of intraday tick-by-tick transaction data of financial assets. The research is therefore of immediate interest in areas of application and will further connect statistics and fields significantly relying on data-analytic tools. In addition, the research will advance mathematical and computational statistics. It will produce doctoral students, who are theoretically and practically versed in both statistics and an area of application. The training and involvement of undergraduate students is also included through regular coursework, independent study and projects. This research concerns the development of a comprehensive framework for the analysis of nonlinear and possibly non-Gaussian functional time series. Such functions naturally arise in a variety of contexts such as the modeling of cumulative intraday returns of financial assets. The research aims at providing a statistical foundation to anal