Our results are as follows. First, we characterize the invariant measures of the dynamics in interest. The bulk of our effort is devoted to the characterization of those of them that are not translation invariant; we call them phase separating invariant measures. In the case of BA, such measures are concentrated on the configurations consisting of two converging infinite blocks of (not necessarily adjacent) particles. In the case of CA 184, a phase separating measure describes the transition from free traffic phase to jammed phase. We also analyze domains of attraction of invariant measures and rates of convergence to them. This analysis then allows us to express the long time limit of particle current in CA 184 as a function of certain characteristics of its initial distribution, when it is translation invariant. This expression has been used in a companion paper (V. Belitsky, J. Krug, E. J. Neves and G. Schütz, A cellular automaton model for two-lane traffic, J. Stat. phys.103(5/6):945--971 (2001)) to show the enhancement of cars current caused by the possibility of lane changes in a model of traffic on a two-lane highway that was created by putting two CA 184s in parallel. Our other results concern hydrodynamic limits of BA and CA 184. We prove that if the integrated profile of initial particle configuration of BA or CA 184 converges, as n , to some stochastic process W(x), x R, when being re-scaled by n ^-1 along x-axis and by c _n ^-1 along y-axis for some sequence c _n , then the integrated profile of particle configuration at time n under the same re-scaling, will converge, as time , to the local moving minimum of the process W(·), that is, to the process W ^min(·) defined by W ^min(x):=min{W(y) :x-1 y x+1}. This hydrodynamic limit is then interpreted in terms of the limiting shape of surface in SG.