How small is the smallest period of a closed trajectory of a Reeb flow? In this talk I will present recent answers to instances of this question in three-dimensions which reveal connections between systolic and symplectic geometry. I will present results both of a positive and of a negative nature. Namely, in some situations there are sharp bounds for the systolic ratio, which is defined as the ratio between the square of the smallest period and the contact volume, while in other situations the systolic ratio is unbounded. Our results confirm a conjecture of Babenko and Balacheff and disprove a conjecture of Hutchings. All this is joint work with Abbondandolo, Bramham and Salomão.