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Sunlight looks white, but it holds all the colours of the rainbow. As it passes through the air, blue gets bounced around far more than the other colours. So it reaches your eyes from every part of the sky, and the sky looks blue.

White sunlight is a mix of colours, and each colour has its own wavelength: short for blue, long for red. The molecules in the air deflect short wavelengths much more. This is called scattering: blue light, scattered in every direction, fills the sky.

It is Rayleigh scattering. Nitrogen and oxygen molecules are far smaller than the wavelength of light, and they scatter it more strongly the shorter the wave: blue (around 450 nm) much more than red (around 700 nm). Violet would scatter even more, but the Sun emits less of it and our eyes are not very sensitive to it. At sunset the light crosses much more air, loses its blue on the way, and the reds remain.

For scatterers much smaller than λ, the Rayleigh cross-section scales as λ⁻⁴: between 450 and 700 nm, blue is scattered about 5.9 times more than red. The sky's spectrum is the solar spectrum weighted by λ⁻⁴ and by the optical depth along the line of sight; at sunset the longer optical path depletes the blue. Scattered light is also polarised, most strongly at 90° from the Sun — which is why a polarising filter darkens the sky.

In the dipole limit the per-molecule cross-section is σ(λ) = (8π/3)·k⁴·|α|², with k = 2π/λ, corrected for the dispersion of the refractive index and for the King factor (≈1.05 for air), which accounts for molecular anisotropy. The scattered spectrum rises towards short wavelengths, but the perceived hue comes from integrating it against the CIE colour-matching functions: the eye's low sensitivity below 450 nm shifts it from violet to blue. Radiative-transfer models add multiple scattering and Mie extinction by aerosols, which desaturates the sky towards the horizon.