Abstract: In this survey article we first briefly introduce some concepts related to the period function of a planar smooth (or analytic) vector field, and its isochronicity, monotonicity, and the number of critical periods. Then, we introduce some important results in this field, especially about the isochronous centers, the period functions associated to the elliptic and hyperelliptic Hamiltonian functions, and the period functions of quadratic integrable systems. Besides these results we list some conjectures and problems in Section 6, which may provide topics for further studies.

Key words: Period function, Isochronicity, Monotonicity, Critical period