Infinite families of curves are constructed of genus 2 and 3 over Q whose jacobians have high rank over Q. More precisely, if [Escr ] is an elliptic curve with rank at least r over Q, an infinite family of curves are constructed of genus 2 whose jacobians have rank at least r+4 over Q, and, under certain conditions, an infinite family of curves are constructed of genus 3 whose jacobians have rank at least 2r over Q. On specialisation, a family of curves are obtained of genus 2 whose jacobians have rank at least 27 and a family of curves are obtained of genus 3 whose jacobians have rank at least 26; one of these has rank at least 42.