Severe cuts are a possibility. Most commonly either from scalpels, from sharp tools such as chisels or from broken glassware.
Once again, prevention is the best approach. Sharp tools should only be used, carefully, by those who are trained to do so.
In the case of a serious cut, minimize the bleeding by raising the affected part, getting gravity to help slow the blood flow. If the cut is on the hand or arm, the victim can use the back of a chair or something else to help support the raised arm. If it is in the leg, lower the casualty to the floor and raise the wound as high as possible, again using a chair can be helpful.
You need to reduce blood loss as far as possible. Do not try a tourniquet – these can cause permanent damage to blood vessels or nerves. If possible get the casualty to apply pressure on or as close to the cut as possible, using fingers, a pad of cloth or better, a sterile dressing. If blood seeps through the dressing, do not remove it, just apply another one on top. (Don’t use paper towel – it will disintegrate and stick to the wound) If the casualty is unable to do so, apply pressure yourself, but protect yourself from contamination by blood if at all possible.
Leave any embedded large bodies: by being in place they can sometimes help to slow the rate of blood loss. Just press around them.