This talk explores the relationship between 3-dimensional lens spaces and smooth 4-manifolds that bound them under various topological constraints—topics that connect to several central conjectures in low-dimensional topology. After reviewing the classifications of Lisca, Greene, and Aceto–McCoy–JH Park, I will present recent joint work with Wookhyeok Jo and Jongil Park investigating which lens spaces can bound smooth 4-manifolds with second Betti number one. In particular, we exhibit infinite families of lens spaces that bound simply connected 4-manifolds with b₂ = 1, yet do not bound 4-manifolds consisting of a single 0-handle and 2-handle. Moreover, we construct infinite families of lens spaces that bound 4-manifolds with b₁ = 0 and b₂ = 1, but do not bound simply connected 4-manifolds with b₂ = 1. These constructions are motivated by the study of rational homology projective planes with cyclic quotient singularities.