The goal of this thesis is to develop generalisations of the Car-Oriented Mean-Field approximation for the Totally Asymmetric Simple Exclusion Process (TASEP) in the inhomogeneous case, that is, under Open Boundary Conditions (OBC) and inhomogeneous hopping rates. We investigate the non-equilibrium stationary-state probability distribution of the model through its marginals in various settings. Since the main source of inspiration for these generalisations is the Cluster approximation, we first study several Cluster approximations (namely, the Mean-Field, Pair, and Triplet approximations) under OBC. We compare them with some kinetic Monte-Carlo simulations both in the case of homogeneous inner hopping rates and in the case of sitewise disorder.