Let $A$ and $B$ be two permutations in $\text{Sym}(n)$ that ``almost commute'' -- are they a small deformation of permutations that truly commute? More generally, if $R$ is a system of words-equations in variables $X = \{x_1, \ldots ,x_d\}$ and $A_1, \ldots, A_d$ are permutations that are nearly solutions; are they near true solutions?

It turns out that the answer to this question depends only on the group presented by the generators $X$ and relations $R$. This leads to the notions of ``stable groups'' and ``testable groups''.

We will present a few results and methods which were developed in recent years to check whether a group is stable or testable. We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic, and with invariant random subgroups.