The purpose of this paper is to construct a new purely algebraic theory of theta functions over a field of characteristic \(p\geq 0\). Let A be an abelian variety over the field \({\mathbb{C}}\) of complex numbers, g a (meromorphic) theta function belonging to A, and \(x_ i\) \((i=1,2,3)\) the coordinate variables on 3 copies of the universal covering space \(V_ A\) of A. If we put \((*)\quad F(x_ 1,x_ 2,x_ 3)=g(x_ 1+x_ 2+x_ 3)g(x_ 1)g(x_ 2)g(x_ 3)/g(x_ 1+x_ 2)g(x_ 2+x_ 3\quad)g(x_ 3+x_ 1),\) then F is a rational function on \(A\times A\times A\), which is determined (up to a constant factor) by the divisor X of g on A. On the other hand even in the case of characteristic p\(>0\), given an abelian variety A and a divisor X on it, a function F on \(A\times A\times A\) is defined by X in the above sense. On the authors' standpoint the problem is: for A find a k(A)-algebra \({\mathcal C}_ A\), functorial with respect to A, such that for each divisor X on A (hence, for F) the equation (*) has a solution g in it. Two kinds of solutions are given to this problem: the theta functions on the Barsotti-Tate group, and those on the Tate space. So far an abelian variety A was given first. In the last two sections 7, 8, abstract theta functions are discussed in the above two ways and using them the authors construct an (abelian) variety.