Several inches of snow is likely to fall before it changes to sleet and freezing rain, mainly along and west of Interstate 95, which runs down the US east coast.
Moderate to locally heavy snow is forecast across the Central Plains and Upper Midwest Tuesday into Wednesday.
And more is expected across the Cascades and Northern Rocky Mountains by the middle of the week.
Winter storm warnings are in place for these areas, where snow has the potential to fall up to 8 inches on higher ground, the National Weather Service said.
As much as a quarter to half an inch of ice could form in some places, according to AccuWeather.
Flights could be delayed or cancelled at major airports such as Chicago, Washington DC, New York and Boston because of aircraft de-icing and low visibility.
School closures across the mid-Atlantic are also likely on Wednesday.
The next round of snow should begin arriving to southern Utah and northern Arizona by Thursday.
Elsewhere, very low temperatures are predicted from Montana to Minnesota with highs in the teens and low 20s Fahrenheit (below 0C).
Wind chill alerts are in effect for some areas.