Fluxonium superconducting circuits were originally proposed to realize highly coherent qubits. In this talk, we explain how these circuits can be used to implement and harness qutrits, by tuning their energy levels and matrix elements via an external flux bias. In particular, we investigate arrays of fluxonium qutrits, and their potential for the quantum simulation of exotic quantum matter. We identify different operational regimes, classified according to the plasmon-like versus fluxon-like nature of the qutrit excitations. Highly tunable on-site interactions are complemented by correlated single-particle hopping, pair hopping and non-local interactions, which naturally emerge and have different weights in the different regimes. We investigate the rich ground-state phase diagram of qutrit arrays and propose practical dynamical experiments to probe the different regimes. Future directions towards analogue simulation of exotic quantum matter are presented.