Topological properties of Delaunay triangulations
Yes. See

Dillencourt, Michael B.; Smith, Warren D. Graph-theoretical conditions for inscribability and Delaunay realizability. Discrete Math. 161 (1996), no. 1-3, 63–77. 

However, if a topological triangulation has no separating triangles, and no chords connecting pairs of vertices on its outer face, then it is always realizable as a Delaunay triangulation.