In this talk we give an algebraic proof of the fact that any Weil number is the Frobenius of an abelian variety over a finite field (joint work with Ching-Li Chai). We recall a (well-known) easy proof of the theorem proved by Weil who showed that the Frobenius of an abelian variety over a field with q elements is a Weil q-number (the first proven case of the Weil conjectures). We discuss the fact that Weil numbers are easily classified and constructed. The Honda-Tate theory classifies (using the result by Weil) isogeny classes of abelian varieties over a finite field with the help of Weil numbers. These concepts will be defined and several proofs will be given.