is a proper smooth surface over a finite field F,. The primary purpose of this paper is to develop arithmetic theory of the Brauer group Br(K) of K. Here we have to assume that the ring J(X, (x) of the regular functions on X has no embedding into R. In general all results hold true modulo 2-torsion. Its description depends on the algebraic K-theory on X. The fundamental tool with which we play our game is the reciprocity map from Br(K) to the character group of the idele class group of X which has an explicit presentation by symbols. As another essential ingredient, we use the concept of arithmetic cohomology of X initiated by Lichtenbaum [L-2]. We can say that one of the key points of the construction lies in a linkage between the K-theory of X and its arithmetic cohomology. This point of view will become more conspicuous in the paper [Sa-5], where more general cases are treated, including the higher dimensional case.