  We present an algorithm solving the following problem: given two genus 2
curves over a field k with isogenous Jacobians, compute such an isogeny
explicitly. This isogeny can be either an l-isogeny or, in the real
multiplication case, an isogeny with cyclic kernel; we require that k have
large enough characteristic and that the curves be sufficiently generic. Our
algorithm uses modular equations for these isogeny types, and makes essential
use of an explicit Kodaira--Spencer isomorphism in genus 2.
