This survey is concerned with  positive solutions of
nonlinear parabolic equations. Assuming that the underlying domain and
the equation have certain reflectional symmetries, the presented
results show how positive solutions reflect the symmetries. Depending
on the class of solutions considered, the symmetries for all times or
asymptotic symmetries are established. Several classes of problems,
including fully nonlinear equations on  bounded domains, quasilinear
equations on $\R^N$, asymptotically symmetric equations, and cooperative
parabolic systems, are examined from this point of view.
Applications of the symmetry results in the study
of asymptotic temporal behavior of solutions are also shown.