  The optimal weak transport problem has recently been introduced by Gozlan
et.\ al. We provide general existence and duality results for these problems on
arbitrary Polish spaces, as well as a necessary and sufficient optimality
criterion in the spirit of cyclical monotonicity. As an application we extend
the Brenier-Strassen Theorem of Gozlan-Juillet to general probability measures
on $R^d$ under minimal assumptions. A driving idea behind our proofs is to
consider the set of transport plans with a new (`adapted') topology which seems
better suited for the weak transport problem and allows to carry out arguments
which are close to the proofs in the classical setup.
