In 2013 and 2014 a revolution took place in the understanding of the discrete logarithm problem (DLP) in finite fields of small characteristic. Consequently, many cryptosystems based on cryptographic pairings were rendered completely insecure, which serves as a valuable reminder that long-studied so-called hard problems may turn out to be far easier than initially believed. In this article, Robert Granger gives an overview of the surprisingly simple ideas behind some of the breakthroughs and the many computational records that have so far resulted from them.