  We show explicit estimates on the number of $q$--rational points of an
$F_q$--definable affine absolutely irreducible variety of the algebraic closure
of the finite field $F_q$ of $q$ elements. Our estimates for a hypersurface
significantly improve previous estimates of W. Schmidt and M.-D. Huang and
Y.-C. Wong, while in the case of a variety our estimates improve those of S.
Ghorpade and G. Lachaud in several important cases. Our proofs rely on
elementary methods of effective elimination theory and suitable effective
versions of the first Bertini theorem.
