There are different species of scorpions; some are mild, others can be deadly. In the US, the smaller the scorpion, the more deadly. The worst in the US is the Arizona bark scorpion; however, it generally does not call for emergency care except for the very young or the very old. Scorpion venom is a neurotoxin. The specific species will determine how the sting site reacts, but many do not cause swelling (although it FEELS like it is massively swollen). The pain can be excrutiating and lasts several days. As with all stings, there is a risk of anaphylaxis. And yes, they can and will sting multiple times.
The "one" time I was stung, it got me on both arms, two or three stings on one, one on the other. It was on the refrigerator, and when I opened the door it threw it onto me, my jumping threw it to the other arm. And this was all in the dark--I now turn on lots of lights if I am walking through the house at night.