  The theory of digital sequences is a fundamental topic in QMC theory. Digital
sequences are prototypes of sequences with low discrepancy. First examples were
given by Il'ya Meerovich Sobol' and by Henri Faure with their famous
constructions. The unifying theory was developed later by Harald Niederreiter.
Nowadays there is a magnitude of examples of digital sequences and it is
classical knowledge that the star discrepancy of the initial $N$ elements of
such sequences can achieve a rate of order $(\log N)^s/N$, where $s$ denotes
the dimension. On the other hand, very little has been known about the $L_p$
norm of the discrepancy function of digital sequences for finite $p$, apart
from evident estimates in terms of star discrepancy. In this article we give a
review of some recent results on various types of discrepancy of digital
sequences. This comprises: star discrepancy and weighted star discrepancy,
$L_p$-discrepancy, discrepancy with respect to bounded mean oscillation and
exponential Orlicz norms, as well as Sobolev, Besov and Triebel-Lizorkin norms
with dominating mixed smoothness.
