<p>Go-or-grow approaches represent a specific class of mathematical models used to describe populations where individuals either migrate or reproduce, but not both simultaneously. These models have a wide range of applications in biology and medicine, chiefly among those the modeling of brain cancer spread. The analysis of go-or-grow models has inspired new mathematics, and it is the purpose of this review to highlight interesting and challenging mathematical properties of reaction–diffusion models of the go-or-grow type. We provide a detailed review of biological and medical applications before focusing on key results concerning solution existence and uniqueness, pattern formation, critical domain size problems, and traveling waves. We present new results related to the critical domain size and traveling wave problems, and we connect these findings to the existing literature. Moreover, we demonstrate the high level of instability inherent in go-or-grow models. We argue that there is currently no accurate numerical solver for these models, and emphasize that special care must be taken when dealing with the “monster on a leash”.</p>