Planarity of the XC(=)NHY linkage has been investigated in unprecedented detail in a number of relatively simple compounds, including formamide (X = Y = H), acetamide (X = CH3, Y = H), urea (X = NH2, Y = H), carbamic acid (X = OH, Y = H), and methyl carbamate (X = OCH3, Y = H). Reliable estimates of the equilibrium structures of formamide, cyanamide, acetamide, urea, carbamic acid, methylamine, dimethyl ether, and methyl carbamate are derived, mostly for the first time. It is shown that formamide, considered prototypical for the amide linkage, is not typical as it has a planar equilibrium amide linkage corresponding to a single-minimum inversion potential around N. In contrast, several molecules containing the CONH linkage seem to have a pyramidalized nitrogen at equilibrium and a double-minimum inversion potential with a very small inversion barrier allowing for an effectively planar ground-state structure. Observables of rotational spectroscopy, including ground-state inertial defects, quadrupole coupling and centrifugal distortion constants, and dipole moment components, as well as equilibrium CO and C-N bond lengths are reviewed in their ability to indicate the planarity of the effective and possibly the equilibrium structures.