Predicting the quantum behaviour of strongly-interacting electrons remains at the forefront of contemporary theoretical and computational physics. When the density matrix renormalisation group algorithm was formulated in 1992, it was quickly adopted as the go-to approach for simulating ground states of one-dimensional quantum lattice systems. Its reformulation in terms of tensor networks paved the way for innovative computational methods for targeting finite-temperature physics and extracting dynamical correlation functions that can directly be probed in spectroscopy experiments. Tensor networks have also been generalised to two-dimensional lattices, known as projected entangled-pair states (PEPS), but have not yet witnessed an equally widespread adoption. Nonetheless, PEPS methods have been delivering state-of-the-art ground state results in the past few years, exploring systems that are inaccessible to competing methods. Particular emphasis is on the Hubbard model and its extensions, which are relevant for high-temperature cuprate superconductors and recently engineered bilayer materials. This project will further develop the PEPS toolkit to simulate extended Hubbard models at finite temperature, in order to characterise their intricate phase diagram and obtain new perspectives on the physical mechanism that drives these phases and their transitions. In addition, novel strategies will be explored to extract dynamical information, that can benefit the whole community.