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We consider semilinear parabolic equations ut=uxx+f(u) on R. We give an overview of results on the large time behavior of bounded  solutions, focusing in particular on their limit profiles as t with respect to the locally uniform convergence. The collection of such limit profiles, or, the ω-limit set of the solution, always contains a steady state.  Questions of interest then are whether---or under what conditions---the ω-limit set consists of steady states, or even a single steady state. We give several theorems and  examples pertinent to these questions.