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 $\mathbb R^N$, 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.