The 2001 outbreaks of foot-and-mouth disease (FMD) in Europe and concern about the possibility of an intentional introduction of a devastating foreign animal disease triggered renewed interest in both theoretical and experimental research related to FMD. Theoretical models of disease transmission, which influenced the tactical decisions of the United Kingdom (UK) government during the epidemic, resulted in large numbers of uninfected animals being slaughtered. After the epidemic, the adopted control policies were sharply criticized. Still, the role of computational modeling for analyzing the scope of the epidemic and devising control strategies was recognized as substantial and necessary. The goal of this paper is to review the entire range of FMD modeling efforts undertaken since 1990. We have three specific purposes: 1. To evaluate the relative strengths and weaknesses of each model and of the types of modeling approaches. 2. To highlight shortcomings that could be overcome through novel computational modeling efforts; 3. To identify new directions in quantitative research that would contribute to optimizing methods for livestock disease control.