How robust are classical properties of random walks? In particular, if the trajectory of a random walk undergoes equilibrium dynamics in the space of walks, does behavior occur which has zero probability under random walk measure? We introduce dynamical random walks (DRW) and provide some answers to this question. We also discuss how the DRW gives the "coin tosser" a means to study an infinite-dimensional diffusion. (Joint work with D. Khoshnevisan and P. Mendez.)