Translational tilings: structured or wild?

The study of the structure of translational tilings has captivated mathematicians, scientists, and the general public for centuries and continues to thrive at the crossroads of analysis, combinatorics, dynamics, logic, number theory, and geometry. This vibrant field seeks to uncover the delicate divide between rigid structures and unpredictable, ``wild'' behaviors that arise when sets fill space by translations without gaps or overlaps. We provide an overview of this study and recent developments, highlighting its multidisciplinary nature and offering a glimpse into the process behind the results.