I never get tired of the blue sky.
Vincent Van Gogh

So why is the sky blue? Time for a trip back to science class: it's all about the colors in visible light. Light might appear white, but it's made up of the colors of the rainbow. Most of these colors pass through the atmosphere, but blue light, being made up of short, small waves, can't make it, and is scattered. The result is the blue color we see.

The angle of the sun affects this color, as well, and is why the sunrise and sunset results in a differently colored sky. When the sun rises or sets, it's at a lower angle, and red light (which is much longer than blue light) passes through. The blue light is further scattered, and the result is a mix of red, orange, and yellow along with the blue.

Further reading:
NASA | UCR | HowStuffWorks | Wikipedia

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