\begin{alignat}{3}
u_t &= Lu + f(t,x,u,\nabla u), &\   &x\in \Omega,&\            &t>0,\\
Bu&=0,&                                            &x\in\partial \Omega,& &t>0.
\end{alignat}

Here $\Omega$ is  a domain in $R^N$, $N \ge  1$, $L$ is a second
order elliptic operator, $f$ is   a  real-valued function,   and $B$   is  a
boundary operator of a standard form (Dirichlet, Neumann or Robin). We
impose appropriate regularity  hypotheses  on the above functions  and
domain  and  always assume  that  $f$  is   periodic in $t$  (this  in
particular  includes autonomous   equations).  We  survey results  and
techniques  in the   study  of the   asymptotic  behavior of   bounded
solutions. The following topics are discussed:

a) The comparison principle and monotone dynamical systems
(convergence or asymptotic periodicity of typical trajectories).

b) One space dimension (convergence and Poincar\'e-Bendixson
theorems, Floquet bundles and perturbations).

c) Positive solutions on higher-dimensional symmetric domains
(asymptotic symmetrization, spatio-temporal asymptotics).

d) Equations with a gradient structure (convergence theorems
via analyticity or normal hyperbolicity).

e) Realization of vector fields on invariant manifolds (existence
of chaotic  dynamics; existence of trajectories with
high-dimensional limit sets; semilinear  heat equations  with
nonconvergent bounded solutions).