  The extent to which a sequence of finite length differs from a shifted
version of itself is measured by its aperiodic autocorrelations. Of particular
interest are sequences whose entries are 1 or -1, called binary sequences, and
sequences whose entries are complex numbers of unit magnitude, called
unimodular sequences. Since the 1950s, there is sustained interest in sequences
with small aperiodic autocorrelations relative to the sequence length. One of
the main motivations is that a sequence with small aperiodic autocorrelations
is intrinsically suited for the separation of signals from noise, and therefore
has natural applications in digital communications. This survey reviews the
state of knowledge concerning the two central problems in this area: How small
can the aperiodic autocorrelations of a binary or a unimodular sequence
collectively be and how can we efficiently find the best such sequences? Since
the analysis and construction of sequences with small aperiodic
autocorrelations is closely tied to the (often much easier) analysis of
periodic autocorrelation properties, several fundamental results on
corresponding problems in the periodic setting are also reviewed.
