This note is primarily concerned with symmetry properties of solutions
of parabolic equations. To put the questions in broader perspective,
we first recall several  symmetry results on  elliptic equations
(on bounded and unbounded domains). We then proceed by summarizing
earlier theorems on asymptotic symmetry for parabolic equations
on bounded domains. Finally, we announce a new theorem on
nonautonomous equations on $R^N$ which asserts that positive
solutions decaying at spatial infinity are asymptotically radially
symmetric about some center. As our discussion of the symmetry
problem reveals, dealing with  parabolic equations on $R^N$,
one is faced with interesting extra difficulties not present  
in elliptic equations or in  parabolic equations on bounded domains.