It has been known for some years that there are deep connections between continued fractions and integrable systems: one of the earliest examples appears in Moser's work on solutions of the finite Kac-van Moerbeke (or Volterra) lattice, but there are many other examples e.g. in recurrence coefficients for orthogonal polynomials arising in random matrix theory, which satisfy (discrete and continuous) Painleve equations. In this talk we describe our recent work on continued fractions of Jacobi type (J-fractions) for a certain family of functions on hyperelliptic curves, based on a construction of van der Poorten related to Somos-4 sequences (corresponding to genus g=1). We explain how to interpret van der Poorten's result for all genera g, in terms of a family of discrete integrable systems. This not only leads to an elementary derivation of Hankel determinant formulae for Somos-4 found by Chang, Hu & Xin, but also provides a natural construction of higher genus analogues of Chebyshev polynomials, and produces genus g solutions of the infinite Toda lattice. The J-fractions naturally arise from even models of hyperelliptic curves, but recent work with John Roberts and Pol Vanhaecke also reveals another family of discrete integrable systems associated with continued fraction of Stieltjes-type (S-fractions) and odd models of the same curves, which yield solutions of the infinite Volterra lattice. In particular, we find that the g=2 S-fraction corresponds to an integrable map with 2 degrees of freedom, discovered in a recent classification of 4D maps with Lagrangian structure by Gubbiotti, Joshi, Viallet & Tran. Other 4D integrable maps found by the latter authors turn out to be connected with the modified Volterra lattice: our additional joint work with Federico Zullo has revealed that they arise from the same genus 2 S-fraction, but as BTs in the sense of Sklyanin.