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Online popularity has enormous impact on opinions, culture, policy, and profits, especially with the advent of the social Web and Web advertising. Yet the processes that drive popularity in our online world have only begun to be explored. We provide a quantitative, large scale, longitudinal analysis of the dynamics of online content popularity in two massive model systems, the Wikipedia and an entire country's Web space. In these systems, we track the change in the number of links to pages, and the number of times these pages are visited. We find that these changes occur in bursts, whose magnitude and time separation are very broadly distributed. This finding is in contrast with previous reports about news-driven content, and has profound implications for understanding collective attention phenomena in general, and Web trends in particular. To make sense of these empirical results, we offer a simple model that mimics the exogenous shifts of user attention and the ensuing non-linear perturbations in popularity rankings. While established models based on preferential attachment are insufficient to explain the observed dynamics, our stylized model is successful in recovering the key features observed in the empirical analysis of our systems.