In this thesis, we consider stationary random mosaics with a focus on the Poisson-Voronoi mosaic and the Poisson-Delaunay mosaic. We consider properties of stationary random mosaics in R2, such as mean value results of the typical cell. Further, we simulate various mean value results of the typical cell, a random neighbor of the typical cell, and the zero cell for the Poisson-Voronoi mosaic in R2. Some theory of point processes is introduced that is needed for random mosaics, including Palm theory, marked point processes, and the Pois point process. Finally, we consider an incremental flip-based algorithm for generating the Voronoi mosaic.