  This research monograph focuses on the arithmetic, over number fields, of
surfaces fibred into curves of genus 1 over the projective line, and of
intersections of two quadrics in projective space. The first half takes up and
develops further the technique initiated by Swinnerton-Dyer in 1993, and later
generalised by Colliot-Th\'el\`ene, Skorobogatov and Swinnerton-Dyer, for
studying rational points on pencils of curves of genus 1. The second half,
which builds upon the first, is devoted to quartic del Pezzo surfaces and to
higher-dimensional intersections of two quadrics. Conditionally on two
well-known conjectures (Schinzel's hypothesis and the finiteness of
Tate-Shafarevich groups of elliptic curves), it establishes the Hasse principle
for all smooth n-dimensional intersections of two quadrics with n>2 as well as
for a large class of del Pezzo surfaces of degree 4.
