In this talk I will survey some results on the existence of periodic points of symplectomorphisms defined on closed orientable surfaces of positive genus g. Namely, I will describe some symplectic flows on such surfaces possessing finitely many periodic points and describe a non-Hamiltonian variant of the Hofer-Zehnder conjecture for symplectomorphisms defined on surfaces; this conjecture provides a quantitative threshold on the number of fixed points (possibly counted homologically) which forces the existence of infinitely many periodic points. This is joint work in progress with Marcelo Atallah and Brayan Ferreira.