TITLE: Existence of zero cycles of degree one vs existence of rational points
QUESTION [26 upvotes]: Let $k$ be a field (I'm mainly interested in the case where $k$ is a number field, however results for other fields would be interesting), and $X$ a smooth projective variety over $k$. 
By a zero cycle on $X$ over $k$ I mean a formal sum of finitely many (geometric) points on $X$, which is fixed under the action of the absolute Galois group of $k$. We can define the degree of a zero cycle to be the sum of the multiplicities of the points. 
Now, if $X$ contains a $k$-rational point then it is clear that $X$ contains a zero cycle of degree one over $k$. 
What is known in general about the converse? That is, which classes of varieties are known to satisfy the property that the existence of a zero cycle of degree one over $k$ implies the existence of a $k$-rational point? For example what about rational varieties and abelian varieties?
As motivation I shall briefly mention that the case of curves is easy. Since here zero cycles are the same as divisors we can use Riemann-Roch to show that the converse result holds if the genus of the curve is zero or one, and there are plently of counter-examples for curves of higher genus. However in higher dimensions this kind of cohomological argument seems to fail as we don't (to my knowledge) have such tools available to us.

REPLY [7 votes]: On the positive side (with no restriction on the field), I don't think anyone has mentioned quadrics. For a smooth quadric, the existence of a 0-cycle of degree one is equivalent to the existence of a point with odd degree. This implies the existence of a rational point, by a theorem of Springer.