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The gonality sequence $(d_r)_{r\geq1}$ of a smooth algebraic curve comprisesthe minimal degrees $d_r$ of linear systems of rank $r$. We explain twoapproaches to compute the gonality sequence of smooth curves in $\mathbb{P}^1\times \mathbb{P}^1$: a tropical and a classical approach. The tropicalapproach uses the recently developed Brill--Noether theory on tropical curvesand Baker's specialization of linear systems from curves to metric graphs. Theclassical one extends the work of Hartshorne on plane curves to curves on$\mathbb{P}^1 \times \mathbb{P}^1$.

Brill-Noether theory of curves on $\mathbb{P}^1 \times \mathbb{P}^1$: tropical and classical approach