  In this paper, we systematically summarize and enhance the understanding of
weak convergence and functional limits of record numbers in discrete-time
random walks under Spitzer's condition, and extend these findings to
$\sigma$--record numbers using similar methods. Additionally, we identify a
sufficient condition for the existence of functional limits for record numbers
in continuous-time random walks. Finally, we derive corresponding results for
large deviations, moderate deviations, and laws of the iterated logarithm
pertaining to record numbers in discrete-time random walks.
