We consider the Cauchy problem
\begin{alignat*}{2}
    & u_t=u_{xx}+f(u), &\qquad & x\in \mathbb R,\  t>0,\\
    & u(x,0) =u_0(x), && x\in \mathbb R,
\end{alignat*}
where $f$ is a locally Lipschitz function on $\mathbb R$  with  $f(0)=0$, and $u_0$ is a nonnegative function in $C_0(\mathbb R)$, the space of continuous functions with limits at $\pm\infty$ equal to 0.  Assuming that the solution $u$ is bounded, we study its large-time behavior from several points of view. One of our main results is a general quasiconvergence theorem saying that all limit profiles of $u(\cdot,t)$ in $L^\infty_{loc}(\mathbb R)$ are steady states. We also prove convergence results under additional conditions on $u_0$. In the bistable case, we characterize the solutions on the threshold between decay to zero and  propagation to a positive steady state, and show that the threshold is sharp for each increasing family of initial data in $C_0(\mathbb R)$.