Controlled radical polymerization (CRP) can be viewed as the middle ground between living anionic polymerization (LAP) and conventional free radical polymerization (FRP). It combines the precise control over polymer structure offered by LAP, under a tolerant reaction condition similar to FRP. One of the most studied CRP is atom transfer radical polymerization (ATRP), with over 10,000 papers published since its introduction in 1995. Despite the numerous studies, knowledge on its fundamental mechanism is still lacking, as evident from the lack of expression for full MWD and polydispersity that account for termination reaction. Since termination is unavoidable in ATRP, the existing expressions give inaccurate predictions as dead chains accumulate. In this study, we derived expressions for full MWD at low conversion and for polydispersity. These expressions allow us to quantify and gain better understanding on the contribution of termination. In addition, the resulting polydispersity expression shows better agreement than the existing equation when correlated with experiment data. In addition to the aforementioned questions, there are also controversies regarding the kinetics of surface-initiated ATRP, with researchers divided into two schools of theories. We evaluated the validity of these theories by comparing their predictions to experimental trends. Both theories were found to be inadequate in explaining all the experimental observations, thus triggering an investigation of the graft density. Graft density is an important determining property for polymer brushes, yet little is known about what affects its final value. Through simulations, we investigated the effect of experiment factors on the grafting density. A decrease in the amount of deactivator is found to decrease the grafting density, which could be explained by an increase in the number of monomers added per activation cycle. This knowledge allows us to explain the conflicting experiment observations regarding the growth trends of polymer layers reported in the literatures. / Thesis / Doctor of Philosophy (PhD) / Polymer materials are used almost everywhere in our daily life from clothing to water bottle. This wide range of applications owes to the nearly infinite possible properties that polymer can possess. Different polymerization processes to synthesize polymers have their own weaknesses and strengths. Herein we investigated the fundamental mechanism of one of the currently most attractive polymerization systems, atom transfer radical polymerization (ATRP). This process allows the synthesis of polymers with precisely tailored chain microstructures, making it possible to create polymer with sophisticated properties. Using modeling approaches, we derived explicit expressions for determining chain properties, allowing detailed investigation of how various factors affect these properties. Through these investigations, we obtained better understanding on the mechanism of ATRP in solution and on surface. This knowledge is crucial in providing insight and guiding experimental designs for better control over the material properties.