Certain singularly perturbed partial differential equations exhibit a phenomenon known as dynamic metastability, whereby reaction-diffusion patterns evolve asymptotically exponentially slowly in time. For instance, this metastability occurs in the propagation of thin interfaces for phase separation models, including the Cahn-Hilliard equation, which has applications to material science. It also occurs for the Gierer-Meinhardt reaction-diffusion system that models pattern formation in morphogenesis. The speaker will illustrate metastable behavior for certain classes of partial differential equations of reaction-diffusion type and will show how asymptotic, spectral and numerical analysis can be used to obtain a precise characterization of the slow dynamics. A necessary condition for metastability in these diverse types of reaction-diffusion equations is that the spectrum associated with the linearization of the partial differential equation around a certain robust canonical solution, such as a standing wave, contains exponentially small eigenvalues. As will be shown, this condition is by no means sufficient for the existence of metastable behavior. The talk will be aimed towards a broad audience including those who have interest in partial differential equations, motion by curvature, dynamical systems, and numerical analysis.