Graphical designs are subsets of vertices that can perfectly average sufficiently smooth functions on a graph; they are quadrature rules on graphs analogous to spherical designs. Graphical designs have multiple applications – to sampling and random walks for example, and connect to multiple areas of mathematics such as polyhedral (convex) geometry, combinatorics and representation theory. After introducing the basic definitions, I will explain recent results with Zawad Chowdury and Stefan Steinerberger that showcase the power of designs to find well-known combinatorial structures in highly structured graphs. Examples include orthogonal arrays in hypercube graphs, combinatorial block designs in the Johnson graph and t-wise permutations in the transposition graph on the symmetric group.