  We prove a general inequality for estimating the number of points of
arbitrary complete intersections over a finite field. This extends a result of
Deligne for nonsingular complete intersections. For normal complete
intersections, this inequality generalizes also the classical Lang-Weil
inequality. Moreover, we prove the Lang-Weil inequality for affine as well as
projective varieties with an explicit description and a bound for the constant
appearing therein. We also prove a conjecture of Lang and Weil concerning the
Picard varieties and \'etale cohomology spaces of projective varieties. The
general inequality for complete intersections may be viewed as a more precise
version of the estimates given by Hooley and Katz. The proof is primarily based
on a suitable generalization of the Weak Lefschetz Theorem to singular
varieties together with some Bertini-type arguments and the
Grothendieck-Lefschetz Trace Formula. We also describe some auxiliary results
concerning the \'etale cohomology spaces and Betti numbers of projective
varieties over finite fields and a conjecture along with some partial results
concerning the number of points of projective algebraic sets over finite
fields.
