We introduce a new definition of bubbles in discrete-time models based on the discounted stock price losing mass at some finite drawdown under an equivalent martingale measure. We provide equivalent probabilistic characterisations of this definition and give examples of discrete-time martingales that are bubbles and those that are not. In the Markovian case, we provide sufficient analytic conditions for the presence of bubbles. We also show that the existence of bubbles is directly linked to the existence of a non-trivial solution to a linear Volterra integral equation of the second kind involving the Markov kernel. Finally, we show that our definition of bubbles in discrete time is consistent with the strict local martingale definition of bubbles in continuous time in the sense that a properly discretised strict local martingale in continuous time is a bubble in discrete time. The talk is based on joint work with Dörte Kreher.