Quantum field theory (QFT) is a conceptual framework for understanding the behaviour of subatomic particles—the most successful, mathematically rigorous formulation of QFT is in the language of operator algebras. In this thesis, we describe the construction of specific kinds of QFTs using operator-algebraic methods. Once we have described their construction in detail, we use tensor network methods (which are at the centre of modern quantum physics) to build approximations of these QFTs. We finish with a discussion on the relationship between our tensor networks and those used in toy models of the AdS/CFT correspondence.