The Kitaev honeycomb model (KHM) attracts several venues of theoretical research due to its relevance in topological quantum computation, integrable models, and characterization of novel frustrated magnets. The KHM is exactly solvable for S = 1/2, was studied with numerical methods for S = 1, and displays a large-S solution capturing its physics for S > 3/2. The S = 3/2 KHM is unique among the spin-S Kitaev models due to a massive ground-state quasidegeneracy that hampered previous numerical and analytical studies. In this seminar, we will discuss the ground state properties of the S = 3/2 KHM as uncovered by a combination of state-of-the-art DMRG simulations and SO(6) Majorana parton mean-field theory [Jin et al., Nat. Commun. 13, 3813 (2022); Natori et al., PRB 108 075111]. In particular, we show how extensions of the KHM onto Kugel-Khomskii (KK) models are helpful to explain anomalous features of the S = 3/2 isotropic KHM, such as its flat bands and its susceptibility to strong first-order quantum phase transitions. The remarkable quantitative agreement between numerical and analytical results strongly indicates that we found an excellent theoretical description of this model, fostering studies on the domain of validity of the mean-field approximation. We show preliminary results of a study on the mixed-spin KHM with S = 1/2 and S = 3/2 species. For this model, the mean-field theory indicates a three-fold degenerate ground state. Such states are observed as low-energy eigenstates in exact diagonalization, and DMRG studies of the model find different ground states with the same degeneracy. Further extensions of this work on KK models implementable in current quantum simulators are discussed by the end of the talk.