The unique chemical and physical properties of fullerenes are to a great extent determined by their unusual curved shape. One of the most active fields in fullerene chemistry is the search for molecular receptors able to form stable associates with them in solution, with the purification from fullerite and the self-assembly of nanoscale electronic devices as main driving forces. In this tutorial review we present a brief overview of the recent advances in the construction of molecular receptors that feature curved concave surfaces complementary to the convex surface of fullerenes as a key recognizing element.