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