  This is a survey on the Darboux theory of integrability for polynomial vector
fields, first in $\R^n$ and second in the $n$-dimensional sphere $\sss^n$. We
also provide new results about the maximum number of parallels and meridians
that a polynomial vector field $\X$ on $\sss^n$ can have in function of its
degree. These results in some sense extend the known result on the maximum
number of hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in
function of the degree of $\Y$.
