Abstract: A new conforming Delaunay triangulation is presented in two-dimensional domains with curved boundaries and inner curve restrictions.Approximated segments of curved boundaries and inner curve restrictions are defined.A new conforming Delaunay refinement method is presented and the convergence of this algorithm is proved.The presentation of curved boundaries and inner curve restrictions in the generated mesh is implemented.The uniform topology of approximated segments and primitive curves is proved.Some examples that generating guaranteed-quality triangulation meshes with curved boundaries and inner curve restrictions are presented.