What causes flat red circles on skin that do not itch but turn brown over time?

The condition you are describing might be called "solar lentigines", also known as sun spots, age spots, or liver spots. They are flat, round, dark spots on the skin that are usually brown or black. They are caused by exposure to ultraviolet radiation (UV) from the sun or artificial sources, such as tanning beds. UV radiation can damage the skin and cause the formation of melanin, the pigment that gives skin its color. Solar lentigines are usually harmless and do not require treatment, but they can be cosmetically unappealing to some people.