  If $V$ is an irreducible algebraic variety over a number field $K$, and $L$
is a field containing $K$, we say that $V$ is diophantine-stable for $L/K$ if
$V(L) = V(K)$. We prove that if $V$ is either a simple abelian variety, or a
curve of genus at least one, then under mild hypotheses there is a set $S$ of
rational primes with positive density such that for every $\ell \in S$ and
every $n \ge 1$, there are infinitely many cyclic extensions $L/K$ of degree
$\ell^n$ for which $V$ is diophantine-stable. We use this result to study the
collection of finite extensions of $K$ generated by points in $V(\bar{K})$.
