Most of the mesh refinement algorithms focus on producing mesh with quality triangles. In our approach, we try to refine a Delaunay mesh such that the final mesh is stable. Normally, we refine a mesh by inserting a Steiner point into it. However, whenever we insert a new point, the connectivity of the node around candidate node may change. Older edges may disappear and new edges may appear in the mesh. In our approach we formalize the concept of Roaming Region to find out a region where the movement of the candidate node will not change the connectivity of the existing nodes. After formalizing the notion of Roaming Region, we propose two algorithms to relocate the candidate node to a more stable location in the mesh i.e., towards the center of Roaming region. The pdf below describes our approach of Stable Mesh Refinement.