We establish a priori bounds for positive solutions of semilinear elliptic systems of the form {-Delta u = g(x, u, v) in Omega, -Delta v = f(x, u, v) 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 on u and v. Based on these bounds, existence of positive solutions is proved.417796