Functional data analysis is a burgeoning area in statistics. However, much of the literature to date deals primarily with methods for collections of independent functional observations, which are not well suited to application with time series of curves. In this paper, a functional time series model is proposed for the purpose of curve forecasting. The model is a synthesis of ideas stemming from traditional functional data analysis and from dynamic factor analysis. The primary contribution of the model is that it accounts for both smooth functional behavior and dynamic correlation in time series of curves. Specifically, it is hypothesized that observed data represents a discrete sampling of an underlying smooth time series of curves. These curves themselves are functions of unobserved dynamic factors with corresponding factor loadings that take the form of a functional curve; the model is thusly named the functional dynamic factor model (FDFM). Based on distributional assumptions regarding the observed data and unobserved factors, maximum likelihood estimation is proposed. To ensure that the estimated factor loading curves do represent smooth curves, roughness penalties are added to the likelihood, resulting in a penalized likelihood. The unobserved time series factors are considered as a problem of missing data for which the Expectation Maximization algorithm (EM) is well suited as the tool of estimation. As part of the EM, generalized cross validation (GCV) is used to select the optimal smoothing parameter corresponding to each smooth factor loading curve. As an iterative estimation procedure, the EM in this context can be computationally intensive. To this end, several computational efficiencies are derived to expedite estimation. Model performance is illustrated through simulation and through rather varied applications, including industrial, climatological and financial settings. Based on the simulation studies, the FDFM results in accurate parameter estimation as compared to those of benchmark models. For both simulated and applied data, forecast results for the FDFM are comparable to results from other models used in their respective applied area. Finally, several extensions of the functional dynamic factor model and areas of future research are described.