In this section, we focus mainly on Gauss, Jacobi, and Kloosterman sums over finite fields, with brief mention of Eisenstein and Jacobsthal sums. Throughout, Fq is a finite field of characteristic p with q = pr elements. (In Subsection 6.1.3, the exponent r will be taken to be the order of p (mod k) for a fixed integer k.) We refer to [240] for proofs of many of the results in this section. In some cases, the proofs need modification because of differing definitions of the trivial character χ0: in Definition 6.1.1 below, χ0(0) = 0, while in [240], χ0(0) = 1.