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Mars is one of the most intriguing planets in the solar system. It is the fourth terrestrial planet and is differentiated into a core, mantle and crust. The crust of Mars is divided into the Southern highlands and the Northern lowlands. The largest volcano in the solar system, Olympus Mons is found on the crustal dichotomy boundary. The presence of isolated volcanism on the surface suggests the importance of internal activity on the planet. In addition to volcanism in the past, there has been evidence of present day volcanic activity. Convective upwelling, including decompression melting, has remained an important contributing factor in melting history of the planet. In this thesis, I investigate the production of melt in the mantle for a Newtonian rheology, and compare it with the melt needed to create Tharsis. In addition to the melt production, I analyze the 3D structure of the mantle for a stagnant lithosphere. I vary different parameters in the Martian mantle to understand the production of low or high degree structures early on to explain the crustal dichotomy. This isothermal structure in the mantle contributes to the geoid and topography on the planet. I also analyze how much of the internal density contributes to the surface topography and areoid of Mars. In contrast to Mars, Ceres is a dwarf planet in the Asteroid belt. Ceres is an icy body and it is unclear if it is differentiated into a core, mantle and crust yet. However, studies show that it is most likely a differentiated body and the mantle consists of ice and silicate. The presence of brucite and serpentine on the surface suggests the presence of internal activity. Being a massive body and also believed to have existed since the beginning of the solar system, studying Ceres will shed light on the conditions of the early solar system. Ceres has been of great interest in the scientific community and its importance has motivated NASA to launch a mission, Dawn, to study the planet. Dawn will collect data from the dwarf planet when it arrives in 2015. In my modeling studies, I implement a similar technique on Ceres, as followed on Mars, and focus on the mantle convection process and the geoid and topography. The silicate-ice mixture in the mantle gives rise to a non-Newtonian rheology that depends on the grain size of the ice particle. The geoid and topography observed for different differentiated scenarios in my modeling can be compared with the data from the Dawn mission when it arrives at Ceres in 2015. / Ph. D.