This workshop, sponsored by AIM and the NSF, will focus on the geometry of Hilbert schemes. Despite their importance, many of their geometric properties remain mysterious. Classically, the Hilbert scheme of projective space was shown to be path-connected, while the Hilbert scheme of points on a smooth curve or surface is smooth. Beyond these foundational results, most research has centred on constructing pathological examples of Hilbert schemes that parametrize curves and higher-dimensional objects, revealing a striking contrast between the well-behaved and pathological cases. Recent breakthroughs have uncovered new smooth components of Hilbert schemes, identified extreme singularities, and exposed unexpected connections to combinatorics and linkage theory. This workshop will bring together experts and early-career researchers from around the world to advance our understanding of Hilbert schemes.