In the paper we find conditions on the pair (ω1, ω2) which ensure the boundedness of the maximal operator and the Calderón-Zygmund singular integral operators from one generalized Morrey space Mp,ω1 to another Mp,ω2 , 1 < p < ∞, and from the space M1,ω1 to the weak space WM1,ω2 . As applications, we get some estimates for uniformly elliptic operators on generalized Morrey spaces.