Zeta or L - Functions Are Modelled on the Riemann'szeta Function Originally Defined By the Seriesand Then Extended to the Whole Complexplane. the Zeta Function Has an "Euler Product", a "Functionalequation" and Though Very Much Studied Still Keeps Secret Many of ItsproperTies, the Greatest Mystery Being the So-Called Hiemann Hypothesis. Manysimilar (Or Thought to Be Similar) Serieshave Been Introduced In Arithmetic,Algebraic Geometry and Even Topology, Dynamics (We Won't Discuss the Latter).We Plan Basically to Discuss Zeta Functions Attached to Algebraic Varietiesover Finite Fields and Global Fields. the First Applications of Zeta Functions Haw Been Thearithmetic Progession Theorem (Dirichlet. 1837) "There Exists One (Hence Infinitely) Prime Congruent to a Modulo B.Whenever a and B Are Coprime" and the Prime Number TheOrem(Kiemann 1859, With an Incomplete Proof: Hadamard and De La Vallee Poussin,1896) "The Number of Primes Lessthan X Is Asymptotic to Butfurther Applications Were Not Restricted to the Study of Prime Numbers, Theyinclude the Study of the Ring of Algebraic Integers, Class Field Theory, Theestimation of the Size of Solutions of (Some) Diophantine Equations, Etc.Moreoverhave Provided or Suggested Fundamentallinks Between Algebraic Varieties (Motives Over Q). Galois Representations,Modular or Au- Tomorphic Forms: For Example, Though They Do Not Appearexplicitly In Wiles Work, It Seems Fair to Say They Played an Important Role Inthe Theory That Finally Led to the ...