Do coloured objects form coloured shadows?

Shadows are simply an area that does not get as much light as the areas around it. That being said, if the object is totally opaque (or you can't see through it), then the shadow will simply be darker (usually a shade of black) because of the lack of light. If the object is transparent, on the other hand, then what light does go through will tint the light that does find it's way through the object.