Computing obstructions is a useful tool for determining the dimension and singularity of a Hilbert scheme at a given point. However, this task can be quite challenging when the obstruction space is nonzero. In a previous joint work with S. Mukai and its sequels, we developed techniques to compute obstructions to deforming curves on a threefold, under the assumption that the curves lie on a "good" surface (e.g., del Pezzo, K3, Enriques, etc.) contained in the threefold. In this talk, I will review some known results in the case where the intermediate surface is a K3 surface and the ambient threefold is Fano. Finally, I will discuss the deformations of certain space curves lying on a complete intersection K3 surface, and the construction of a generically non-reduced component of the Hilbert scheme of P^5.