In this talk, we present the following three classical and unresolved problems of low-dimensional topology. They are about objects called homology 3-spheres first introduced by Henri Poincaré at the beginning of the twentieth century.

1. Which homology 3-spheres bound contractible 4-manifolds or homology 4-balls? (main problem)
2. Which homology 3-spheres are homology cobordant to $S^3$?
3. Which homology 3-spheres embed in $S^4$?

We discuss interactions amongst these three problems from a historical perspective and state several open problems. Addressing the plumbing construction of manifolds, we introduce natural sources of homology 3-spheres including Brieskorn spheres and Seifert fibered homology spheres. In the end, we list all currently known results for plumbed homology 3-spheres.