We report a new mechanism for creating vortices in a class of flows that are linearly stable and believed, by most researchers, to be also finite-amplitude stable. We find that the vortices form in numerical simulations of stably-stratified Couette flows (both plane and circular), as well as in simulations of protoplanetary disks around forming protostars. Our study was motivated by the fact that protoplanetary disks must have flow instabilities to form stars. The mechanism that we discovered allows small-amplitude perturbations (i.e., with small volumes and Rossby numbers) to form vortices that are large in volume and amplitude (with a Rossby number of order unity). The energy of the vortices becomes large, and it is supplied the kinetic energy of the background shear flow. The underlying mathematics of the finite-amplitude instability lies in Math 53 and 54. Our vortices have an unusual property: a vortex that grows from a single, local perturbation triggers a new generation of vortices to grow at nearby locations. After the second generation of vortices grows large, it triggers a third generation. The triggering of subsequent generations continues ad infinitum so that a front dividing the vortex-dominated flow from the unperturbed flow advances until the entire domain fills with large vortices. The vortices do not advect across the region, the front of the vortex-populated fluid does. The region in protoplanetary disks where we have found this new mechanism is thought to be stable; thus, in the astrophysical literature this region is called the dead zone. Because the vortices we report here arise in the dead zone, grow large, and spawn new generations of vortices that march across the domain, we refer to them as zombie vortices. We consider the mechanism of the zombie vortices’ growth and advance in a proposed lab experiment: circular Couette flow with a vertically stably-stratified Boussinesq fluid (i.e., salt water) with a density that is linear with height. Because this flow is nearly homogenous, the first vortex formed by the initial instability self-replicates in an approximately spatially self-similar manner and fills the domain with a lattice of 3D vortices, which persists, despite the fact that the flow is turbulent.