If \(G = \langle X\rangle\) is a finitely generated group, the growth function \(\gamma_G(n)\) is the number of elements of \(G\) word length at most \(n\). We ignore the fine-scale detail of this function and focus on how fast it tends to infinity. The growth of a group is a fundamental quasi-isometric invariant, but there are still many mysteries. It can be as slow as polynomial (e.g., for nilpotent groups), or as fast as exponential (e.g., for free groups), and nothing else was known until the 80's when Grigorchuk gave his famous example of a group of intermediate growth, i.e., neither polynomial nor exponential, and it is now known (Erschler--Zheng, 2020) that this group has growth roughly \(\exp(n^{0.767})\). Grigorchuk's "gap conjecture" predicts that there is some constant \(c > 0\) such if the growth is slower than \(\exp(n^c)\) then it should be polynomial (which is equivalent to virtual nilpotence, by a theorem of a Gromov). This is known for residually nilpotent groups with \(c = 1/2\), and Wilson (2011) showed that it holds for residually soluble groups with \(c = 1/6\). Elena Maini and I have now improved this to \(c = 1/4\) in the residually soluble case. To be precise, if \(G\) is residually soluble and its growth is \(< \exp(\frac{n^{1/4}}{100})\) for large \(n\) then \(G\) is in fact virtually nilpotent. In this talk I will give an overview of this landsacpe, including a basic introduction to the theory of growth, and by the end of the talk I will give the whole proof of a slightly weaker bound with exponent \(\frac{1}{4.16}\).