Dynamical systems arising in engineering and science are often subject to random influences (noise). The noisy processes may be Gaussian or non-Gaussian, which are modeled by Brownian motion or -stable Levy motion, respectively. Non-Gaussianity of the noise manifests as nonlocality at a macroscopic level. Stochastic partial or ordinary differential equations (SPDEs or SDEs) with Brownian motion or Levy motion are appropriate models for these systems. To understand dynamics under uncertainty, topological, geometric and analytical approaches are taken to examine the quantities (e.g., escape probability, Conley index) that carry dynamical information, and the structures (e.g., invariant manifolds and foliations, slow manifolds and slow foliations) that act as dynamical building blocks. The speaker will first present an overview of recent research on random invariant manifolds, foliations and their approximations, and then focus on understanding random dynamics by examining escape probability, highlighting its application with prototypical examples in biophysical and physical settings.