We introduce a notion of smooth fields of operators following the notion of smooth fields of Hilbert spaces defined by L. Lempert and R. Szoke. We show that, if V is the connection of a smooth field of Hilbert spaces, then del = [del ,center dot ] defines a connection on a suitable space of fields of operators. We prove a smooth version of the reduction theorem and we apply it to show that, if h(q,p) = IIpII2 and {u, h} = 0 then Weyl quantization maps u into an operator admitting a decomposition as a smooth field of operators over the interval (0, infinity). Moreover, we prove an explicit formula to compute the derivatives of those fields of operators. We also introduce a notion of smooth field of C* -algebras and we provide an explicit example.