A lot depends on what you want to know about realizability. You could argue that Thom's paper settles the problem: in the mod 2 case, every homology class is realizable by a map; in the integral case, some integer multiple of every homology class is realizable by a map.

There has been recent work on trying to bound the integer we need to multiply by in the integral case. See for example the preprint https://arxiv.org/abs/2303.10240 by Vasilii Rozhdestvenskii, to appear in Journal of Topology.

For realizability by embeddings, Thom says a lot. There are some nice examples of non-realizable homology classes due to Bohr, Hanke and Kotschick in

Bohr, Christian; Hanke, Bernhard; Kotschick, Dieter, Cycles, submanifolds, and structures on normal bundles, Manuscr. Math. 108, No. 4, 483-494 (2002). ZBL1009.57043.

Note however that that paper contains a mis-statement in Remark 2 at the bottom of page 485, where it is claimed that collapse at the $E^2$ page of the Atiyah-Hirzebruch spectral sequence for oriented bordism implies that all integral homology is realizable by "immersed submanifolds". All we can really conclude is that everything is realized by a map, as in Steenrod's problem.

The situation for realizability by immersions is somewhat more subtle. András Szűcs and I gave the first examples of mod 2 homology classes not realizable by immersions in

Grant, Mark; Szűcs, András, On realizing homology classes by maps of restricted complexity, Bull. Lond. Math. Soc. 45, No. 2, 329-340 (2013). ZBL1270.57068.

We also have some results about non-realizability by maps with some presecribed set of (multi-)singularities, in the above paper and in

Grant, Mark; Szűcs, András, Homologies are infinitely complex, Topol. Methods Nonlinear Anal. 45, No. 1, 55-61 (2015). ZBL1368.57013.

Zhenhua Liu asked here on MathOverflow the very interesting (in my view) question of whether there are integral homology classes realizable by immersions but not embeddings: Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds

A lot of these questions can be approched using classical (but nevertheless difficult) obstruction theory.

A lot depends on what you want to know about realizability. You could argue that Thom's paper settles the problem: in the mod 2 case, every homology class is realizable by a map; in the integral case, some integer multiple of every homology class is realizable by a map.

There has been recent work on trying to bound the integer we need to multiply by in the integral case. See for example the preprint A lower bound in the problem of realization of cycles by Vasilii Rozhdestvenskii, to appear in Journal of Topology.

For realizability by embeddings, Thom says a lot. There are some nice examples of non-realizable homology classes due to Bohr, Hanke and Kotschick in

Bohr, Christian; Hanke, Bernhard; Kotschick, Dieter, Cycles, submanifolds, and structures on normal bundles, Manuscr. Math. 108, No. 4, 483–494 (2002). ZBL1009.57043.

Note however that that paper contains a mis-statement in Remark 2 at the bottom of page 485, where it is claimed that collapse at the $E^2$ page of the Atiyah–Hirzebruch spectral sequence for oriented bordism implies that all integral homology is realizable by "immersed submanifolds". All we can really conclude is that everything is realized by a map, as in Steenrod's problem.

The situation for realizability by immersions is somewhat more subtle. András Szűcs and I gave the first examples of mod 2 homology classes not realizable by immersions in

Grant, Mark; Szűcs, András, On realizing homology classes by maps of restricted complexity, Bull. Lond. Math. Soc. 45, No. 2, 329–340 (2013). ZBL1270.57068.

We also have some results about non-realizability by maps with some presecribed set of (multi-)singularities, in the above paper and in

Grant, Mark; Szűcs, András, Homologies are infinitely complex, Topol. Methods Nonlinear Anal. 45, No. 1, 55–61 (2015). ZBL1368.57013.

Zhenhua Liu asked here on MathOverflow the very interesting (in my view) question of whether there are integral homology classes realizable by immersions but not embeddings: Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds

A lot of these questions can be approched using classical (but nevertheless difficult) obstruction theory.