With the increased deployment of renewable energy sources and digitization, the evolution from the traditional grid to the smart grid has become an urgent priority. As the grid becomes increasingly complex with increased functionalities and embedded intelligence, its stability and resilience become essential to provide secure and dependable services to the end-user. Moreover, the smart grid operations incorporate the end-users in demand response and EV management, thus giving rise to the concept of prosumers. Previous studies have focused primarily on a single aspect of the smart grid. However, to realize the futuristic grid vision, a fundamental study is required to analyze operations while incorporating most aspects. The comprehensive analysis must be performed at generation, transmission and distribution levels spanning geographical regions and time scales. This thesis provides a framework for this analysis by studying the system stability and resilience at various time and spatial scales. Firstly, grid stability has been analyzed at a smaller scale by framing it as a frequency control problem while incorporating the cyber-physical aspects with uncontrollable and controllable decentralized energy sources. A framework has also been developed to study the grid resilience to communication packet drop rates and cyber-attacks. Then, the role of consumer response and market price elasticity has been explored in relation to grid stability. Secondly, the operations of the energy storage devices have been analyzed with non-parametric test statistic, with hourly generation scheduling under stochastic wind and contingency scenarios. Thirdly, the grid stability and resilience analysis are performed at a larger scale by statistical and machine learning methods. The aim was to determine the relationship between the topological features and nodal voltage stability index for various power networks. The eigen-spectrum of the power networks has been utilized along with real-time voltage measurement using spectral filters to find a critical community of nodes to improve the power network resilience to nodal attacks. Hence, the work in this thesis provides a generalized analysis framework at all levels of smart grids utilizing the concept of control theory, optimization and data science for stability and resilience.