Abstract: A finitely generated group G is said to have uniform exponential growth if the number of elements in G that are represented by words of length at most n over some generating set of G is bounded below by a single exponential function in n that does not depend on the generating set. In 1981, Gromov asked whether every group of exponential growth has uniformly exponential growth. While this has been confirmed for many natural classes of groups, the general answer turned out to be negative, as shown by Wilson in 2004. Since the groups constructed by Wilson are not finitely presented, the question remained open whether there exist finitely presented groups of non-uniform exponential growth. Further properties that resisted being combined with non-uniform exponential growth include simplicity, Kazhdan's property (T) and acylindrical hyperbolicity. In this talk I will show that for each of these properties there is a group of non-uniform exponential growth admitting it. This talk is based on joint work with Roman Sauer.