As Arctic sea ice starts to melt in the summer, melt ponds form on its surface and, in a matter of days, cover large portions of the ice. Due to their low reflectivity, melt ponds greatly accelerate ice melt. Despite their importance, they are poorly understood due the many processes that control their evolution, which operate on widely separated length-scales. In this thesis, we use idealized models of melt ponds with a goal to provide a fundamental understanding of their evolution. First, we study the case of late-summer ponds that exist on highly permeable first-year sea ice. Assuming that ice is fully permeable, we show that pond coverage evolution can be approximately determined by solving two uncoupled ordinary differential equations (ODEs) in which the rate of change of pond coverage fraction is a function of itself, of the initial ice surface hypsographic curve, and of average melt rates of different regions of the ice. In this way, we show that it is possible to greatly reduce the complexity of pond evolution on permeable ice and to summarize all of the environmental conditions with only a few aggregate parameters. Second, we show that melt pond geometry on both first and multi-year ice can be accurately captured by a simple geometric model where ponds are represented as voids that surround randomly sized and placed circles that represent snow dunes. There are only two model parameters: the characteristic circle radius and the pond coverage fraction. We set these parameters by matching two correlation functions, which determine the typical pond size and their connectedness, between the model and aerial photographs of melt ponds. With parameters calibrated in this way, we reproduce the previously-observed pond size distribution and fractal dimension as a function of pond size over the entire observational range of more than 6 orders of magnitude. Surprisingly, by further studying the correlation functions, we find that late-summer ponds are organized close to the critical percolation threshold. Moreover, we find that ponds from different years and documented at different locations have very similar typical sizes. Third, we explain the observation we made previously that the ponds are organized close to the percolation threshold. We show that, since ponds drain through large holes, the percolation threshold is an upper bound on pond coverage following pond drainage. Furthermore, because of the universality of systems close to the percolation threshold, we show that the pond fraction as a function of the number of open holes follows a universal curve. This curve governs pond evolution during and after pond drainage, which allows us to formulate an equation for pond coverage evolution that captures the dependence on physical properties of the ice and is supported by observations. Finally, we generalize the void model we developed earlier and show that it accurately captures the pre-melt distribution of snow-depth. We find that the snow depth is distributed according to a Gamma distribution which can be fully characterized by the mean and the variance of snow depth. This allows us to derive an analytical formula for pond evolution during early summer when ice is impermeable. By combining all of our results, we find that nearly the entire pond evolution since the onset of melt can be captured with computationally inexpensive analytical models that do not sacrifice accuracy and reveal relationships between pond evolution and measurable ice parameters that would not be captured using more complex models. These findings have significant potential to improve our parameterizations of sea ice albedo in large-scale climate models, thereby advancing our ability to predict the fate of Arctic sea ice. This work was led, performed, and written by Predrag Popovic under supervision of Dorian Abbot and Mary Silber.