Let $G$ be a connected locally compact group with a left Haar measure $\mu$, and let $A,B \subseteq G$ be nonempty and compact. Assume further that $G$ is unimodular, i.e., $\mu$ is also the right Haar measure; this holds, e.g., when $G$ is compact, a nilpotent Lie group, or a semisimple Lie group. In 1964, Kemperman showed that

$$ \mu(AB) \geq \min \{\mu(A)+\mu(B), \mu(G)\} .$$

The Kemperman inverse problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. I will discuss the recent solution of this problem, highlighting the role played by the theory of approximate groups. (Joint with Jinpeng An, Yifan Jing, and Ruixiang Zhang)