  We study the local-global principle for zero-cycles of degree 1 on certain
varieties fibered over the projective space. Among other applications, we prove
that the Brauer-Manin obstruction is the only obstruction to the Hasse
principle and weak approximation for zero-cycles of degree 1 on
Severi-Brauer-variety bundles or Ch\^atelet-surface bundles over the projective
space.
