An infinite group \(G\) is called just-infinite if all of its proper quotients are finite. Since its introduction by McCarthy in the late 1960's, the class of just-infinite groups received a lot of attention. One reason for the importance of just-infinite groups is that, by Zorn's lemma, every finitely generated, infinite group admits a just-infinite quotient. By a celebrated result of Wilson, the study of just-infinite groups can be reduced to the study of simple groups, branch groups, and hereditarily just-infinite groups, i.e. groups all of whose finite index subgroups are just-infinite. After recalling some background on residually finite groups and profinite completions, I will present an elementary idea that gives rise to constructions of all 3 types of just-infinite groups. As an application, we will discuss the first examples of finitely generated just-infinite groups that are residually finite and have positive rank gradient. In fact we will see that these examples have positive first \(L^2\)-Betti-number. This talk is based on joint work with Steffen Kionke.