A tipping point occurs when adjusting one or more parameters of a system causes it to transition abruptly to a new state. Examples include desertification due to grazing, financial crises, and drastic shifts observed in past climate states. Data from such systems, whether "real world" or model-derived, may provide interesting insight into the system. We review the mechanisms underpinning recent statistical work in predicting "tipping points" in 1D time series. We will then discuss analysis of such data, theoretically of any dimension, from the angle of persistent topology.