I will discuss a recent result on the sum-product problem in finite fields, showing that given a subset $A$ of a finite field of sufficiently large characteristic $p$, if $|A|< p^{26/51}$ then $$\max\{|A+A|, |AA|\}> |A|^{16/13 - o(1)}.$$ This talk is based on a joint work with Sophie Stevens (RICAM).