The discoveries of the Hat and Spectre  — single shapes that can be used to form tilings of the plane, but only can form non-periodic ones — lay to rest the longstanding question of the existence of an “aperiodic monotile” but it remains an open question: How complex can the behavior of a single shape of tile be? Can we even tell whether or not a given shape will tile the plane — is the “monotiling problem” even decidable? We’ll survey the status of several related decision and existence problems, across a range of settings, such as hyperbolic space, groups as graphs, or tilings by a single monotile.