We describe here a novel way of defining Hamiltonians for quantum field theories (QFTs), based on the particle-position representation of the state vector and involving a condition on the state vector that we call an "interior-boundary condition." At least for some QFTs (and, we hope, for many), this approach leads to a well-defined, self-adjoint Hamiltonian without the need for an ultraviolet cut-off or renormalization.