Two presentations for a group G which have the same deficiency are called exotic if the corresponding presentation 2-complexes are not homotopy equivalent. The first examples of exotic presentations were found by Dunwoody and Metzler in the 1970s. A long-standing problem has since been to determine whether there exist exotic presentations with non-minimal deficiency. For many groups G, there is a close relationship between the homotopy types of group presentations for G and stably free ZG-modules, with deficiency corresponding to the rank of the stably free module. However, there has previously been no known example of a non-free stably free ZG-module of rank greater than one. The aim of this talk will be to present the first examples in each case. In particular, for each k at least 2, I will construct non-free stably free ZG-modules of rank k and exotic presentations with deficiency k above the minimal value. I will also present a number of new questions.