An electromagnetic wave propagating through space is always a plane wave. It means that its E and H component alternate in a sinusoidal manner, perpendicular to the axis of propagation. It is a "transversal" wave.
Only close to radiators that generate the wave, the shape of the wavefront runs off a "point", so it is spherical. As it travels farther from the radiator, it flattens in a planar wave.
Some radiators utilize an aperture (waveguide horn, parabolic dish); the wave energy is concentrated in a desired direction (therefore those are directional antennas). The larger the aperture measured in wavelengths, the farther from the antenna the energy beam forms. Until the beam is fully formed, the wavefront is not exactly planar; it has a "transition" form between the spherical and planar, distorted by the beam-forming aperture.
Like in optics, we divide the space in two zones: the "near" zone closer to the radiator, and the "far" zone farther away. In the "far" zone, the wavefront is cleanly planar; such wave we want to be.