Very few deserts are formed solely by a rain shadow effect, because they are also influenced by the high atmospheric pressure. However, the North American deserts are usually called rain shadow deserts. The final type of desert to be mentioned is not near the equator, but instead in a climate that is so cold, that the air can only hold a small amount of moisture. There is very little precipitation, and all the surface water is locked in unusable blocks of ice. These conditions have created the Antarctican and Greenland deserts, which are nothing like the usual picture of arid lands.
Deserts are certainly more diverse than most people suspect.
Southwest Parks and Monuments Association. Long-Term Ecological Research Network Office, University of Washington. How to make a desert. University of Texas at El Paso Centennial Museum's Laboratory for Environmental Biology.