Polycrystalline diamond inserts are disclosed. For example, a polycrystalline diamond insert may comprise a polycrystalline diamond layer affixed to a substrate at an interface. In addition the polycrystalline diamond layer may comprise: an arcuate exterior surface, a first region including a catalyst and a second region from which the catalyst is at least partially removed. Further, the arcuate exterior surface may be defined by a portion of the first region including the catalyst and a portion of the second region from which the catalyst is at least partially removed. In another embodiment, the polycrystalline diamond layer may include a convex exterior surface for contacting a subterranean formation, wherein at least a portion of a catalyst used for forming the polycrystalline diamond layer is removed from a region of the polycrystalline diamond layer. Subterranean drilling tools (e.g., percussive drill bits) including at least one polycrystalline diamond insert are disclosed.