Can we approximate the Hilbert space of a quantum field theory by a finite subspace? In this talk I will summarize my understanding of this question, both in the continuum and on the lattice. The emphasis will be on the problems in matching to the continuum and taking the large volume limit. At the end of the talk, I will discuss work in progress on a new formulation of Hamiltonian truncation in infinite volume that may address some of these problems.