We establish a priori bounds for positive solutions of semilinear elliptic systems of the form { Delta u = g(x, v), in Omega -Delta v = f (x, u), in Omega u > 0, v > 0 in Omega u = v = 0 on partial derivative Omega where Omega is a bounded and smooth domain in R-2. We obtain results concerning such bounds when f and g depend exponentially with respect to u and v.62199213