Using semianalytical models we calculate the energy, effective Bohr radius, and radiative lifetime of neutral excitons confined in CdSe colloidal nanoplatelets (NPLs). The excitonic properties are largely governed by the electron-hole in-plane correlation, which in NPLs is enhanced by the quasi-two-dimensional motion and the dielectric mismatch with the organic environment. In NPLs with lateral size L 20 nm the exciton behavior is essentially that in a quantum well, with super-radiance leading to exciton lifetimes of 1 ps or less, only limited by the NPL area. However, for L < 20 nm excitons enter an intermediate confinement regime, hence departing from the quantum well behavior. In heterostructured NPLs, a different response is observed for core-shell and core-crown configurations. In the former, the strong vertical confinement limits separation of electrons and holes even for type-II band alignment. The exciton behavior is then similar to that in core-only NPL, albeit with weakened dielectric effects. In the latter, charge separation is also inefficient if band alignment is quasi-type-II (e.g., in CdSe/CdS), because electron-hole interaction drives both carriers into the core. However, it becomes very efficient for type-II alignment, for which we predict exciton lifetimes reaching microseconds.