"This thesis represents a collection of papers on numerical modeling of permafrost and seasonally freezing ground dynamics. An important problem in numerical modeling of temperature dynamics in permafrost and seasonally freezing ground is related to parametrization of already existing models. In this thesis, a variation data assimilation technique is presented to find soil properties by minimizing the discrepancy between in-situ measured temperatures and those computed by the models. The iterative minimization starts from an initial approximation of the soil properties that are found by solving a sequence of simple subproblems. In order to compute the discrepancy, the temperature dynamics is simulated by a new implementation of the finite element method applied to the heat equation with phase change. Despite simplifications in soil physics, the presented technique was successfully applied to recover soil properties, such as thermal conductivity, soil porosity, and the unfrozen water content, at several sites in Alaska. The recovered properties are used in discussion on soil freezing/thawing and permafrost dynamics in other parts of this thesis. Another part of this thesis concerns development of a numerical thermo-mechanical model of seasonal soil freezing on the lateral scale of several meters. The presented model explains observed differential frost heave occurring in non-sorted circle ecosystems north of the Brooks Range in the Alaskan tundra. The model takes into account conservation principles for energy, linear momentum and mass of three constituents: liquid water, ice and solid particles. The conservation principles are reduced to a computationally convenient system of coupled equations for temperature, liquid water pressure, porosity, and the velocity of soil particles in a three-dimensional domain with cylindrical symmetry. Despite a simplified rheology, the model simulates the ground surface motion, temperature, and water dynamics in soil and explains dependence of the frost heave on specific environmental properties of the ecosystem. In the final part, simulation of the soil temperature dynamics on the global scale is addressed. General Circulation Models are used to understand and predict future climate change, but most of them do not simulate permafrost dynamics and its potentially critical feedback on climate. In this part, a widely used climate model is evaluated and the simulated temperatures are compared against observations. Based on this comparison, several modifications to the Global Circulation Models are identified to improve the fidelity of permafrost and soil temperature simulations. These modifications include increasing the total soil depth by adding new layers, incorporating a surface organic layer, and modifying the numerical scheme to include unfrozen water dynamics"--Leaves iii-iv.