  We define unbounded twisted complexes and bicomplexes generalising the notion
of a (bounded) twisted complex over a DG category [BK90]. These need to be
considered relative to another DG category $B$ admitting countable direct sums
and shifts. The resulting DG category of unbounded twisted complexes has a
fully faithful convolution functor into Mod-$B$ which filters through $B$ if
the latter admits change of differential. As an application, we rewrite
definitions of $A_{\infty}$-structures in terms of twisted complexes to make
them work in an arbitrary monoidal DG category or a DG bicategory.
