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We prove a general inequality for estimating the number of points ofarbitrary complete intersections over a finite field. This extends a result ofDeligne for nonsingular complete intersections. For normal completeintersections, this inequality generalizes also the classical Lang-Weilinequality. Moreover, we prove the Lang-Weil inequality for affine as well asprojective varieties with an explicit description and a bound for the constantappearing therein. We also prove a conjecture of Lang and Weil concerning thePicard varieties and \'etale cohomology spaces of projective varieties. Thegeneral inequality for complete intersections may be viewed as a more preciseversion of the estimates given by Hooley and Katz. The proof is primarily basedon a suitable generalization of the Weak Lefschetz Theorem to singularvarieties together with some Bertini-type arguments and theGrothendieck-Lefschetz Trace Formula. We also describe some auxiliary resultsconcerning the \'etale cohomology spaces and Betti numbers of projectivevarieties over finite fields and a conjecture along with some partial resultsconcerning the number of points of projective algebraic sets over finitefields.

\'Etale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields