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If the cat belongs to you, you need to take a close look at the reason why you need to rehome the cat. If you have any alternatives to getting rid of your cat, you should consider taking them. For example, many people believe it is unsafe to have a litter tray in the house when they are pregnant and therefore look to rehome a much loved feline companion. This fear arises from the (albeit slim) chance of picking up toxoplasma infection. However, scientists now believe the main route of infection is inadequately cooked meat rather than cat feces. If you wear disposable gloves and a mask, and clear the poop within 24 hours of it being produced the risk of infection is negligible. These simple measures could prevent the need for rehoming. If in doubt speak to your veterinarian who can advise you further on protecting your health and minimizing the risk of infection. For example, a new baby in the house may have you feeling protective and worried about how the cat will react and if the cat will sleep on the baby and smother it. This is extremely unlikely to happen but if it's something that worries you, close the cat out of the room when the baby is alone. Take simple precautions, like keep the litter tray away from the baby and wash your hands after petting the cat.  If you are moving abroad, be aware that a pet passport scheme operates widely now. Make inquiries at the embassy of the country you are moving to and ask them about the necessary protocol to import a cat into their country. Your vet can work closely with you to ensure the relevant vaccinations, blood tests, and paperwork is issued, and it's no big deal. It might be that you are moving within the country and just noticed your rental agreement has a 'no pets' clause. Before rehoming the cat, be honest with the landlord and ask if there is the possibility of making an exception. Some landlords put the clause on as a matter of routine, to protect themselves should a tenant have a destructive pet. Many landlords are open to a discussion about pets. It is especially helpful if you can put together a portfolio of documents that illustrate how well-behaved and well-cared for your pet it. A dossier containing records of vaccination certificates, vet checks, references from previous landlords, pictures of your pet doing cute things, like sleeping, can all go some way to convince a wavering landlord. Indeed, cats are hugely beneficial to a growing child. It's been shown that kids raised in a house with pets are less likely to develop asthma and allergies in later life.  Having a pet can reduce stress in owners of all ages. They can also provide love and companionship to their owners, which can be especially important to people who live alone or are house bound. Don't just abandon your cat and don't take it to a shelter unless you have exhausted all other possibilities. You have the responsibility to find it a safe and loving home where it can thrive. This may take some time and effort but you have that obligation to the cat.
Decide whether you really need to rehome your cat. Consider alternatives to rehoming. Keep in mind the benefits of having a cat. Place your cat in a great new home.