  We review recent advances on the record statistics of strongly correlated
time series, whose entries denote the positions of a random walk or a L\'evy
flight on a line. After a brief survey of the theory of records for independent
and identically distributed random variables, we focus on random walks. During
the last few years, it was indeed realized that random walks are a very useful
"laboratory" to test the effects of correlations on the record statistics. We
start with the simple one-dimensional random walk with symmetric jumps (both
continuous and discrete) and discuss in detail the statistics of the number of
records, as well as of the ages of the records, i.e., the lapses of time
between two successive record breaking events. Then we review the results that
were obtained for a wide variety of random walk models, including random walks
with a linear drift, continuous time random walks, constrained random walks
(like the random walk bridge) and the case of multiple independent random
walkers. Finally, we discuss further observables related to records, like the
record increments, as well as some questions raised by physical applications of
record statistics, like the effects of measurement error and noise.
