Abstract: By additive and multiplicative doubly quasi-periodic analytic functions we mean functions satisfying (1·1) and (1·2) respectively. In case they have only poles as singularities, we call them as quasi-elliptic functions. Similarly we may define sectionally holomorphic and doubly quasi-periodic functions. In this paper, we solve the corresponding doubly quasi-periodic Riemann boundary value problem(0·1) by reducing them to doubly periodic ones which were considered and solved in(2) and(3) both for closed contours and open arcs.