W. H. Meeks has asked the following question: For what $g$ does
every (orientation preserving) periodic automorphism of a closed
orientable surface of genus $g$ have an invariant circle? A variant of this question due to R. D. Edwards
asks for the existence of invariant essential circles. Using a
construction of Meeks we show that the answer to his question is negative
for all but 43 values of $g\leq 10000$, all of which
lie below $g=105$. We then show that the work of S. C. Wang on Edwards'
question generalizes to nonorientable surfaces and automorphisms of odd order. Motivated by this,
we ask for the maximal odd order of a periodic automorphism of
a given nonorientable surface. We obtain a fairly complete
answer to this question and also observe an amusing relation between this
order and Fermat primes.