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This paper develops a framework for studying the popularity dynamics of user-generated videos, presents a characterization of the popularity dynamics, and proposes a model that captures the key properties of these dynamics. We illustrate the biases that may be introduced in the analysis for some choices of the sampling technique used for collecting data; however, sampling from recently-uploaded videos provides a dataset that is seemingly unbiased. Using a dataset that tracks the views to a sample of recently-uploaded YouTube videos over the first eight months of their lifetime, we study the popularity dynamics. We find that the relative popularities of the videos within our dataset are highly non-stationary, owing primarily to large differences in the required time since upload until peak popularity is finally achieved, and secondly to popularity oscillation. We propose a model that can accurately capture the popularity dynamics of collections of recently-uploaded videos as they age, including key measures such as hot set churn statistics, and the evolution of the viewing rate and total views distributions over time.