It's easy to see if your cat or dog is left or right handed because they'll show a preference using one paw to reach out. Or they'll used that paw to wipe something off their nose. Our dog Oscar tilts his head to the right if he's curious about a noise. If he spots something, he "points" with his right paw. So individual dogs and cats can be right or left pawed but over the population, the split is about 50-50.
Humans on the other hand(!) are about 90 per cent right handed. Surprisingly it's possible to infer that this goes back two million years to our distant Neanderthal relatives. Researchers examined the grooves in their teeth caused by their use of toothpicks.

Handedness in other primates is less clear. One study found that chimpanzees will point about two-thirds of the time with their right hands. Another observed captive chimpanzees for 10 months, also finding a preference for right-handed gestures but that may be biased if the chimps learned handedness through imitation.
Another study involving nearly 800 great apes found that gorillas, chimps and bonobos prefer their right hand for tasks requiring one hand to hold an object while the other manipulates it. Orangutans showed a preference for their left hand.
The results for other animals is mixed. Toads prefer their right forelimbs to remove pieces of paper stuck to their face. A study of kangaroos found that they are predominantly left handed. They'd use their right hand to support their body while using their left hand to reach for food.
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There are a number of theories to explain why handedness developed. The simplest is that it stems from culture, that we tend to copy each other. Other theories revolve around specialisation of the brain. It's easier to perform two tasks at the same time if they are governed by opposite sides of the brain.
In humans, the left hemisphere is oriented towards processing skills such as language while the right is more musical. That's a crude simplification and we should be wary of tropes such as a person being "right" or "left" brained.
For snails there's a simpler explanation. A snail that coils to the left will have sad love life when almost all its potential partners coil to the right. Recently, however, researchers found three left-leaning snails by using the #snaillove hashtag. Two of those mated, giving birth to 170 normal, right-handed snails.
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