  Extreme events are an important theme in various areas of science because of
their typically devastating effects on society and their scientific
complexities. The latter is particularly true if the underlying dynamics does
not lead to independent extreme events as often observed in natural systems.
Here, we focus on this case and consider stationary stochastic processes that
are characterized by long-range memory and heavy-tailed distributions, often
called fractional L\'evy noise. While the size distribution of extreme events
is not affected by the long-range memory in the asymptotic limit and remains a
Fr\'echet distribution, there are strong finite-size effects if the memory
leads to persistence in the underlying dynamics. Moreover, we show that this
persistence is also present in the extreme events, which allows one to make a
time-dependent hazard assessment of future extreme events based on events
observed in the past. This has direct applications in the field of space
weather as we discuss specifically for the case of the solar power influx into
the magnetosphere. Finally, we show how the statistics of records, or
record-breaking extreme events, is affected by the presence of long-range
memory.
