While it can be embarrassing, frustrating, and sometimes frightening, for the most part, this type of behavior by toddlers is a normal phase that they all pass through. It’s a part of their growth and development, and usually results from a frustration at not being able to express themselves, whether it is wanting a toy, or wanting your attention.
The phase can hit anywhere from 14 months onwards, but tends to be more noticeable when the child is exposed to others, which could mean in a preschool, or social setting. Even children who are linguistically advanced for their ages, are still children, and will be prone to the same frustrations as others.
The first rule of thumb, is not to overreact. While there are different schools of thought on spanking, this is not a situation where it is appropriate, and can only add fuel to the fire. A time-out is in order, generally about two minutes worth on a chair where they can’t stomp on the floor, or kick anything. This also gives you a chance to calm down. During the time-out, do not speak to them, but do explain when you sit them down, that this time-out is because they have bit/hit/kicked someone and that is not allowed.
When the time is up, explain to them again, that the behavior is not acceptable, because it hurts other people. It’s not of much use to ask them how they would feel if someone bit them, since a toddler is unlikely to be able to relate cause and effect, then apply it to themselves. But a non-confrontational “punishment”, and explanation, tells them what they did wrong, and what will happen if they do it again.
If they go right out and repeat the action, take them back for another time-out. Depending on the age of the child, you can explain the concept of apologies, and why they should make one. Use positive reinforcement by praising them for an apology (even if it comes as a kiss), or for going right out and giving their toy to the child they kicked.