Voronoi tessellations have been helpful in solving applied and theoretical problems in disciplines ranging from astronomy to neurology, and they are becoming increasingly useful in spatial statistics. This dissertation provides new applications, methods, and computational resources relating to Voronoi tessellations. The distributional properties of Voronoi cells arising from a tessellation of Southern California earthquakes is studied. This work offers new methods for characterizing earthquakes, provides one of the first distributional investigations of an empirical cluster process, and offers evidence supporting the use of the tapered Pareto distribution in environmetric modeling. Next, Voronoi-type estimators of spatial intensity are studied. Theoretical bounds for the bias are presented and simulations studies are offered, showing a gamma shaped sampling distribution. Finally, an R package is presented which implements new and existing techniques for the analysis of point patterns using Voronoi tessellations.