  We develop and test cluster approximations, which generalize simple
mean--field by taking into account more and more local correlations, for the
Totally Asymmetric Simple Exclusion Process with open boundaries. We consider
in detail the pair and triplet approximations, discussing the improvements with
respect to mean field in various steady state properties. Moreover, we analyze
the recently discovered dynamical transition, describing how the spectrum of
the relaxation matrix changes at the transition.
