Country risk - the likelihood that a state will weaken or fail - and the methods of assessing it continue to be of serious concern to the international community. Country risk has traditionally been assessed by monitoring economic and financial indicators. However, social media (such as forums, blogs, and websites) are now important transporters of citizens' daily conversations and opinions, and as such may carry discernible indicators of risk, but they have been as yet little-used for this task. The Geopolitical Web project is a research effort with the ultimate goal of developing computational approaches for monitoring public opinion in regions of conflict, assessing country risk indicators in the social media of fragile or weakening states, and correlating these risk signals with commonly accepted quantitative geopolitical risk assessments. This paper presents the initial motivation for this data-driven project, collection procedures adopted, preliminary results of an automated topical analysis of the collection's content, and expected future work. By catching and deciphering possible signals of country risk in social discourse we hope to offer the international community an additional means of assessing the need for intervention in or support for fragile or weakening states.