Abstract. A Hausdorff compactification of a topological space $X$ is a compact Hausdorff space $bX$ which contains $X$ as a dense subspace. If $G$ is a topological group, and $bG$ is a Hausdorff compactification of the space $G$, then the subspace $rG=bG\setminus G$ is called a remainder of $G$. If $G$ is locally compact, then $rG$ is compact. If $G$, in addition, is not compact, then $rG$ can be selected to be a singleton, that is, to be trivial.

We consider a more delicate case of a non-locally compact topological group $G$ and discuss, when some remainder $rG$ of $G$ is metrizable, when some remainder of $G$ is paracompact, when some $rG$ has the Baire property, and some other questions on relationship between topological properties of $G$ and $rG$. In particular, a necessary and sufficient condition, in terms of remainders, for a non-locally compact topological group to be separable and metrizable is provided.

The case of remainders of paratopological groups is also briefly considered. It is shown that some basic results on remainders of topological groups cannot be extended to paratopological groups.

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