Hilbert schemes of smooth surfaces are well-studied objects, however not much is known about Hilbert schemes of higher dimensional varieties. In this talk, we will speak about topological properties of Hilbert schemes of affine spaces. In particular, we will compute the homotopy type of the Hilbert scheme of infinite affine space. Time permitting, we will discuss applications to algebraic K-theory, as well as a generalization of this computation for certain Quot schemes. This is joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt Totaro.