Summary: Let \(G\) be a group and \(\varphi\) be an endomorphism of \(G\). A subgroup \(H\) of \(G\) is called \(\varphi\)-inert if \(H^\varphi cap H\) has finite index in the image \(H^\varphi\). The subgroups that are \(\varphi\)-inert for all inner automorphisms of \(G\) are widely known and studied in the literature, under the name inert subgroups. The related notion of inertial endomorphism, namely an endomorphism \(\varphi\) such that all subgroups of \(G\) are \(\varphi\)-inert, was introduced in [the first and the third author, Rend. Semin. Mat. Univ. Padova 127, 213–233 (2012; Zbl 1254.20041)] and thoroughly studied in [the first and the third author, Ann. Mat. Pura Appl. (4) 195, No. 1, 219–234 (2016; Zbl 1341.20056); Ric. Mat. 63, S103–S115 (2014; Zbl 1311.20053)]. The “dual” notion of fully inert subgroup, namely a subgroup that is \(\varphi\)-inert for all endomorphisms of an abelian group \(A\), was introduced in [the second author et al., J. Group Theory 16, No. 6, 915–939 (2013; Zbl 1292.20062)] and further studied in [A. R. Chekhlov, Math. Notes 101, No. 2, 365–373 (2017; Zbl 1387.20045); translation from Mat. Zametki 101, No. 2, 302–312 (2017); “Fully inert subgroups of completely decomposable finite rank groups and their commensurability ”, Vestn. Tomsk. Gos. Univ., Mat. Mekh. 2016, No. 3(41), 42–50 (2016); B. Goldsmith et al., J. Algebra 419, 332–349 (2014; Zbl 1305.20063)]. The goal of this paper is to give an overview of up-to-date known results, as well as some new ones, and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra. We survey on classical and recent results on groups whose inner automorphisms are inertial. Moreover, we show how inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces, and can be helpful for the computation of the algebraic entropy of continuous endomorphisms.