Given a closed, orientable 3-manifold, it is of great interest but often a difficult (and largely open) problem to determine whether the 3-manifold smoothly embeds in 4-dimensional Euclidean space. On the other hand, under additional geometric considerations coming from symplectic topology, such as hypersurfaces of contact type, the problem becomes tractable and in certain cases a uniform answer is possible. In this talk, I will review these concepts and recent results, and explain how they relate to the various notions of convexity (e.g. holomorphic convexity and rational convexity) in complex geometry. I will also report on recent work in progress that extends our results to symplectic 4-manifolds other than Euclidean space, such as rational complex surfaces.