This  survey is devoted primarily to  second order parabolic equations
of the form
\begin{align} u_t &= Lu + f(t,x,u,\nabla u), \quad x\in \Omega,\ t>0,\\
Bu&=0,\qquad                                            x\in\partial \Omega,\ t>0.
\end{align}
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).