With the integration of renewable and distributed energy resources (DER) and advances in metering infrastructure, power systems are undergoing rapid modernization that brings forward new challenges and possibilities, which call for more advanced learning, analysis, and planning tools. While there are numerous problems present in the modern power grid, in this work, this work has addressed four of the most prominent challenges and has shown that how the new advances in generation and metering can be leveraged to address the challenges that arose by them. With regards to learning in power systems, we first have tackled power distribution system topology identification, since knowing the topology of the power grid is a crucial piece in any meaningful optimization and control task. The topology identification presented in this work is based on the idea of emph{prob-to-learn}, which is perturbing the power grid with small power injections and using the metered response to learn the topology. By using maximum-likelihood estimation, we were able to formulate the topology identification problem as a mixed-integer linear program. We next have tackled the prominent challenge of finding optimal flexibility of aggregators in distribution systems, which is a crucial step in utilizing the capacity of distributed energy resources as well as flexible loads of the distribution systems and to aid transmission systems to be more efficient and reliable. We have shown that the aggregate flexibility of a group of devices with uncertainties and non-convex models can be captured with a quadratic classifier and using that classifier we can design a virtual battery model that best describes the aggregate flexibility. For power system analysis and planning, we have addressed fast probabilistic hosting capacity analysis (PHCA), which is studying how DERs and the intermittency that they bring to the power system can impact the power grid operation in the long term. We have shown that interconnection studies can be sped up by a factor of 20 without losing any accuracy. By formulating a penalized optimal power flow (OPF), we were able to pose PHCA as an instance of multiparametric programming (MPP), and then leveraged the nice properties of MPP to efficiently solve a large number of OPFs. Regarding planning in power systems, we have tackled the problem of strategic investment in energy markets, in which we have utilized the powerful toolbox of multiparametric programming to develop two algorithms for strategic investment. Our MPP-aided grid search algorithm is useful when the investor is only considering a few locations and our MPP-aided gradient descent algorithm is useful for investing in a large number of locations. We next have presented a data-driven approach in finding the flexibility of aggregators in power systems. Finding aggregate flexibility is an important step in utilizing the full potential of smart and controllable loads in the power grid and it's challenging since an aggregator controls a large group of time-coupled devices that operate with non-convex models and are subject to random externalities. We have shown that the aggregate flexibility can be accurately captured with an ellipsoid and then used Farkas' lemma to fit a maximal volume polytope inside the aforementioned ellipsoid. The numerical test showcases that we can capture 10 times the volume that conventional virtual generator models can capture. / Doctor of Philosophy / With the integration of renewable and distributed energy resources (DER) and advances in metering infrastructure, power systems are undergoing rapid modernization that brings forward new challenges and possibilities, which call for more advanced learning, analysis, and planning tools. While there are numerous problems present in the modern power grid, in this work, this work has addressed four of the most prominent challenges and has shown that how the new advances in generation and metering can be leveraged to address the challenges that arose by them. With regards to learning in power systems, we first have tackled power distribution system topology identification, since knowing the topology of the power grid is a crucial piece in any meaningful optimization and control task. We next have tackled the prominent challenge of finding optimal flexibility of aggregators in distribution systems, which is a crucial step in utilizing the capacity of distributed energy resources as well as flexible loads of the distribution systems and to aid transmission systems to be more efficient and reliable. For power system analysis and planning, we have addressed fast probabilistic hosting capacity analysis (PHCA), which is studying how DERs and the intermittency that they bring to the power system can impact the power grid operation in the long term. We have shown that interconnection studies can be sped up by a factor of 20 without losing any accuracy. Regarding planning in power systems, we have tackled the problem of strategic investment in energy markets, in which we have utilized the powerful toolbox of multiparametric programming to develop two algorithms for strategic investment. We next have presented a data-driven approach in finding the flexibility of aggregators in power systems. Finding aggregate flexibility is an important step in utilizing the full potential of smart and controllable loads in the power grid and it's challenging since an aggregator controls a large group of time-coupled devices that operate with non-convex models and are subject to random externalities.