East parts of Arizona can pick up 2-4 inches of rain and Phoenix can get 1.5 inches.
This can cause lightning floods in the desert southwest of more than 10 million people under lightning-flood clocks.
The rain will continue until Friday, bringing relief to many areas affected by drought. Much of California is still "unusually dry," according to the U.S. Drought Monitor.
Dry conditions, low humidity and wind gusts will combine to create an increased risk of fires over the next few days.
Later this week, the storm will move east and carry rain to the east, carrying a rainstorm with rain Rocky cliffs and the Midwest.