  We formalize and study several competing notions of versality for an action
of a linear algebraic group on an algebraic variety X. Our main result is that
these notions of versality are equivalent to various statements concerning
rational points on twisted forms of X (existence of rational points, existence
of a dense set of rational points, etc.) We give applications of this
equivalence in both directions, to study versality of group actions and
rational points on algebraic varieties. We obtain similar results on
p-versality for a prime integer p. An appendix, containing a letter from J.-P.
Serre, puts the notion of versality in a historical perspective.
