Interpolation of abstract Cesaro, Copson and Tandori spaces

We study real and complex interpolation of abstract Ces`aro, Copson and Tandori spaces, including the description of Calder'on-Lozanovski{\v \i} construction for those spaces. The results may be regarded as generalizations of interpolation for Ces`aro spaces $Ce{s}_p(I)$ in the case of real method, but they are new even for $Ce{s}_p(I)$ in the case of complex method. Some results for more general interpolation functors are also presented. The investigations show an interesting phenomenon that there is a big difference between interpolation of Ces`aro function spaces in the cases of finite and infinite interval.