The diagrammatic linear response formalism for the Coulomb drag in two-layer systems is developed. This technique can be used to treat both elastic disorder and intralayer interaction effects. In the absence of intralayer electron-electron correlations we reproduce earlier results, obtained using the kinetic equation and the memory-function formalism. In addition we calculate weak-localization corrections to the drag coefficient and the Hall drag coefficient in a perpendicular magnetic field. As an example of the intralayer interaction effects we consider a situation where one (or both) layers are close to (but above) the superconducting transition temperature. Fluctuation corrections, analogous to the Aslamazov-Larkin corrections, to the drag coefficient are calculated. Although the fluctuation corrections do not enhance the drag coefficient for normal-superconductor systems, a dramatic enhancement is found for superconductor-superconductor structures.