Authors’ abstract: We first consider the problem of defining IFS operators on the space \(\mathcal K_{\mathcal C}\) of non-empty compact and convex subsets of \(\mathbb R^d\). After defining a complete metric on \(\mathcal K_{\mathcal C}\), we construct an IFS operator and show some properties. A notable feature is the definition of a type of weak inner product on \(\mathcal K_{\mathcal C}\). We then define a family of complete metrics on the space of all measurable set-valued functions (with values in \(\mathcal K_{\mathcal C}\)), and extend the weak inner product to this space. Following this, we construct IFS operators on these spaces. We close with a brief discussion of the inverse problem of approximating an arbitrary multifunction by the attractor of an IFS.