Hilbert schemes are the prototypical parameter spaces in algebraic geometry. By definition, there is a bijection between subschemes of a fixed projective space with prescribed Hilbert polynomial and the points in a Hilbert scheme. From a geometric perspective, these parameter spaces provide both an important source of higher-dimensional varieties and fundamental insights into families of projective subschemes. Given the inherent complexity of these objects, combinatorics plays an indispensable role in understanding their structure.

These two lectures examine the geometry of Hilbert schemes highlighting the underlying combinatorial features. Starting from the definition, we discuss the construction of Hilbert schemes and identify the nonempty ones. Interspersed with classic examples and open problems, we then examine various geometric properties including connectedness, irreducibility, and singularities.