  Difference sets have been studied for more than 80 years. Techniques from
algebraic number theory, group theory, finite geometry, and digital
communications engineering have been used to establish constructive and
nonexistence results. We provide a new theoretical approach which dramatically
expands the class of $2$-groups known to contain a difference set, by refining
the concept of covering extended building sets introduced by Davis and Jedwab
in 1997. We then describe how product constructions and other methods can be
used to construct difference sets in some of the remaining $2$-groups. We
announce the completion of ten years of collaborative work to determine
precisely which of the 56,092 nonisomorphic groups of order 256 contain a
difference set. All groups of order 256 not excluded by the two classical
nonexistence criteria are found to contain a difference set, in agreement with
previous findings for groups of order 4, 16, and 64. We provide suggestions for
how the existence question for difference sets in $2$-groups of all orders
might be resolved.
