As an alternative to elliptic curve groups, Koblitz (1989) suggested Jacobians of hyperelliptic curves for use in public-key cryptography. Hyperelliptic curves can achieve the same level of discrete log based security as elliptic curves, whilst offering the potential advantage of being defined over much smaller fields. At present however, elliptic curves still outperform hyperelliptic curves in general, because of the significant difference in the complexity of computing group operations. Indeed, when deriving fast explicit formulas for elliptic curve computations, one is aided by the simple geometric "chord-and-tangent" description. In contrast, Cantor's algorithm for arithmetic in Jacobian groups suffers from more computationally heavy operations, such as Euclid's algorithm for finding the gcd of two polynomials, and the chinese remainder theorem. In this talk we discuss recent results which exploit a chord-and-tangent-like analogue for hyperelliptic curves. We give a simple description of higher genus Jacobian arithmetic and show that for genus 2 curves this gives rise to explicit formulas which are significantly faster than their Cantor-based counterparts. This is joint work with Kristin Lauter.