It depends on the model and lens combination you are using. With a high end DSLR and L series lens, there is built in weather-proofing that should protect it from the snow, though the cold temperatures may cause it to stop working temporarily and any ice forming on it could still be a problem. Certainly it would be necessary to clean it off after use to prevent water from staying on it.
On cheaper kits, there won't be as much, if any, weather sealing provided by the body and lens, so it would be necessary to use something like a rain hood to keep the snow off the camera and keep the moisture out. Again, temperature may still cause issues since it is not designed for too cold of operation, but again, it should be temporary, just the electronics might not work well when too cold.
Some cameras, like most of those from Pentax, are actually designed specifically for the cold and are not only weather sealed but also designed to deal with ice and low temperature. They should have no problem in the snow as long as the instructions are followed (such as not changing lenses in the open or when snow is on the lens or body.)
The safest bet, read your manuals, it should say what the environments are that the camera is designed to operate and survive in. Use it in those environments or provide it with protection to keep its immediate environment within those conditions.