One of the most fruitful approaches to studying quantum field theory (QFT) has been lattice gauge theory (LGT), with Hamiltonian formulations growing in popularity as an approach to the quantum simulation of such systems. In this talk, we will give an introduction to LGTs, explain why tensor networks are a fruitful way to study such systems, and present recent progress in the study of Hamiltonian LGTs using tensor networks. We will show how one can construct "gauge invariant states" (think of states that obey Gauss's Law in electromagnetism), and how to use a variational Monte Carlo approach to find the ground states of Hamiltonian LGTs. This approach can work in an arbitrary number of spacetime dimensions, and can be generalized to work with arbitrary gauge groups, including those which are relevant for studying the standard model. We will discuss recent progress in simulating regimes that are known to suffer from the "sign problem".