In this talk we introduce an analytically tractable model of a limit order book where the dynamics are driven by stochastic fluctuations between supply and demand. The model has a natural interpretation for a highly traded market on short time-scales where there is a separation between the time-scale of trading, represented in the model, and a longer time-scale on which fundamentals change. We describe our main result for the model, which is the existence of an explicit limiting distribution for the highest bid, and for the lowest ask, where the limiting distributions are confined between two thresholds. Fluid limits play an important role in establishing the recurrence properties of the model. We use our model to analyze various high-frequency trading strategies (for example market-making, sniping and mixtures of these), and comment on the Nash equilibria that emerge between high-frequency traders when a market in continuous time is replaced by frequent batch auctions. This is joint work with Elena Yudovina, University of Minnesota.