Rampant integration of renewable resources (e.g., photovoltaic / wind-energy conversion systems) and uncontrollable and flexible loads (e.g., plug-in hybrid electric vehicles) are rapidly transforming power systems. In such environments, quantification of the impacts of parametric and input uncertainty are critical to ensure the dynamic security of next-generation power systems. This task is analytically and computationally challenging since power system models are intrinsically nonlinear. Therefore, analytical methods to quantify the impacts of uncertainty are important for the reliable operations. In this presentation, analytic methods to quantify the impacts of parametric and input uncertainties for two important applications in power systems are proposed: i) uncertainty propagation in power system model; and ii) robust stability assessment of power-system dynamics. For the first application, an optimization-based approach is presented to estimate maximum and minimum bounds on state variables while acknowledging worst-case parametric and input uncertainties in the model. The approach leverages a second-order Taylor-series expansion of the states around a nominal (known) solution. Maximum and minimum bounds are then estimated from either semi-definite relaxation of quadratically-constrained quadratic-programming or alternating direction method of multipliers. For the second application, analytic method to assess power systems stability margins while acknowledging uncertainty is presented within the Lyapunov direct method framework. We present numerical case studies for different test systems to demonstrate the proposed methods and validate the approach.