The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of nn points in~ \Real³\Real^3 with spread D\Delta has complexity O(D³)O(\Delta^3). This bound is tight in the worst case for all D = O(Ön)\Delta = O(\sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of kk-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any nn and D = O(n)\Delta = O(n), we construct a regular triangulation of complexity W(nD)\Omega(n\Delta) whose nn vertices have spread D\Delta.