  Let $X$ be a rationally connected algebraic variety, defined over a number
field $k$. We find a relation between the arithmetic of rational points on $X$
and the arithmetic of zero-cycles. More precisely, we consider the following
statements: (1) the Brauer-Manin obstruction is the only obstruction to weak
approximation for $K$-rational points on $X_K$ for all finite extensions $K/k$;
(2) the Brauer-Manin obstruction is the only obstruction to weak approximation
in some sense that we define for zero-cycles of degree 1 on $X_K$ for all
finite extensions $K/k$; (3) a certain sequence of local-global type for Chow
groups of 0-cycles on $X_K$ is exact for all finite extensions $K/k$. We prove
that (1) implies (2), and that (2) and (3) are equivalent. We also prove a
similar implication for the Hasse principle.
  As an application, we prove the exactness of the sequence mentioned above for
smooth compactifications of certain homogeneous spaces of linear algebraic
groups.
