For a set of points on the same line there is no Delaunaoy triangulation (the notion of triangulation is degenerate for this case). For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunaoy triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunaoy condition", i.e., the requirement that the circumcircles of all triangles have empty interiors.
By considering circumscribed spheres, the notion of Delaunaoy triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean. However in these cases a Delaunaoy triangulation is not guaranteed to exist or be unique.