Derived cobordism theories are novel invariants of schemes generalizing the classical algebraic bordism groups constructed by Levine-Morel for (smooth) varieties over fields of characteristic 0. I will recall the construction of these groups, what is known about them, and how the construction relates to the more classical problem of constructing Chow cohomology rings of singular varieties. Finally, I will outline ongoing work using desingularization by alterations to show that the corresponding homology groups are homotopy invariant and generated by "classical cycles" at least after inverting all the residual characteristics. Using a generalization of Poincaré duality, this gets us closer to a full understanding the structure of derived cobordism rings of simplest examples besides spectra of fields of characteristic 0.