  We show that in analogy to the introduction of Poisson structures twisted by
a closed 3-form by Park and Klimcik-Strobl, the study of three dimensional
sigma models with Wess-Zumino term leads in a likewise way to twisting of
Courant algebroid structures by closed 4-forms H.
  The presentation is kept pedagogical and accessible to physicists as well as
to mathematicians, explaining in detail in particular the interplay of field
transformations in a sigma model with the type of geometrical structures
induced on a target. In fact, as we also show, even if one does not know the
mathematical concept of a Courant algebroid, the study of a rather general
class of 3-dimensional sigma models leads one to that notion by itself.
  Courant algebroids became of relevance for mathematical physics lately from
several perspectives - like for example by means of using generalized complex
structures in String Theory. One may expect that their twisting by the
curvature H of some 3-form Ramond-Ramond gauge field will become of relevance
as well.
