The standard formulation of the "cohomological rigidity problem" in toric topology asks whether the homeomorphism or diffeomorphism type of a smooth toric variety is determined by its integral cohomology ring. While such a statement appears unlikely from a basic algebraic topological perspective, no counterexamples are currently known, and many affirmative results supporting this conjecture have been established during last many years. Given that the cohomology ring is a homotopy invariant, it is in fact more natural to ask classification up to homotopy equivalence first rather than homeomorphism. In this talk, we introduce several partial answers to this homotopy question for certain classes of toric varieties. This talk is based on joint works with X. Fu, T. So, and S. Theriault.