Summary: We study the boundedness properties of pseudo-differential operators \(a(x, D)\) and their adjoints \(a(x, D)^*\) with symbols in a certain vector-valued Besov space on Besov spaces \(B^s_{p, q}\) and Triebel spaces \(F^s_{p, q}\) \((0< p, q\leq \infty)\). Applications are given to multiplication properties of Besov and Triebel spaces. We show that our results are best possible for both pseudo-differential estimates and multiplication. Denoting by \((\cdot,\cdot)\) the duality between Besov and between Triebel spaces we derive general conditions under which \((a(x, D) f, g)= (f, a(x, D)^*g)\) holds. This requires a precise definition of \(a(x, D)f\) and \(a(x, D)^*f\) for \(f\in F^s_{p, q}\) and \(f\in B^s_{p, q}\).