More than 145,000 people are without power in some south-eastern states, and thousands of flights are cancelled.
Virginia, Georgia, and North and South Carolina have all declared states of emergency.
The US National Weather Service (NWS) says more than 1ft (30cm) of snow is expected in some areas.
Snow and ice could result in "dangerous travel, power outages, and tree damage", the NWS warned.
Highway patrols reported hundreds of vehicle accidents, according to the Associated Press news agency.
There were also forecasts of possible coastal flooding in some areas, including in New York City and parts of Connecticut, with warnings that roads and infrastructure could be affected.
In Canada's Ontario province, which shares a border with New York state, officials issued storm warnings on Sunday morning for much of the south. Toronto, the capital of the province and Canada's largest city, is forecast to get seven inches (20cm) of snow.
More than 3,000 flights into and out of the US were cancelled on Sunday, according to the FlightAware data tracking website.
Charlotte Douglas International Airport in North Carolina was among the worst hit, with almost 90% of flights scrapped, and a message on its website urged passengers to check with their airlines before travelling to the airport.
In South Carolina, Governor Henry McMaster called on residents to stay off the roads.
"This is going to be a pretty bad storm in the upper part of the state," he said, adding that it was "good news" that the storm was forecast to hit on a weekend and on a public holiday on Monday, when fewer people would need to travel.
"Hopefully, the storm will underdeliver, but it could overdeliver. We just don't know," Georgia Governor Brian Kemp said as he announced preparations for his state.