Cats are usually castrated from 5 months old. This helps minimise urine marking (especially in the house), roaming and fighting with other cats (and associated risks of abscesses and FIV infection). Female cats are usually spayed from 5 months old. This prevents any unwanted litters of kittens, and also removes the risk of diseases of the ovaries or uterus later in life
Male dogs can be castrated from 6 months old. This helps reduce the risk of roaming, also it reduces the risk of prostate disease, perineal hernias and anal problems. It also stops them becoming surprise fathers, and of course it eliminates the risk of testicular cancer!
Female dogs (bitches) can be spayed from 5 months old, either before or after the first season. If neutering is done before the first season it greatly reduces the risk of getting mammary (breast) cancers later on in life. This risk is still reduced if the bitch is neutered after the first season, but by nowhere near as much. It also removes the risk of pyometra (a potentially life-threatening infection of the uterus, usually seen in older dogs), and or course there will be no seasons to cope with and no unwanted litters of puppies.
Rabbits are neutered from 12 weeks of age. It is important to keep the sexes apart before this is done as they can be fertile from a very young age. In males, neutering will decrease aggression whilst in females it prevents the commonest form of cancer in rabbits, uterine cancer which kills many older rabbits.
Plus, statistically, neutered pets live longer!