In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology
3
3
-spheres and the structure of
Θ
Z
3
\Theta _{\mathbb {Z}}^3
. Finally, we briefly discuss the knot concordance group
C
\mathcal {C}
and the rational homology cobordism group
Θ
Q
3
\Theta _{\mathbb {Q}}^3
, focusing on their algebraic structures, relating them to
Θ
Z
3
\Theta _{\mathbb {Z}}^3
, and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology
3
3
-spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.