According to my web search results, there are at least 21 cat breeds in Australia 

Some of the most popular cat breeds in Australia include the Ragdoll, Bengal, Abyssinian, Burmese, Persian, Russian Blue, and Siamese.

My personal favourite breed of cat is the Ragdoll cat, it is quite a large, affectionate cat that goes with the flow. Ragdolls are ideal lap cats because they simply go limp with pleasure when they are being petted, giving them their name. This Ragdoll breed is one of the largest, but don’t let the size intimidate you. Their big blue eyes and loud, throaty purr let you know they’re just big softies. Ragdolls are notoriously social, making them a good fit for bustling homes with plenty of people and friendly pets.

Some interesting facts about Ragdolls include that they all have beautiful blue eyes. Ragdoll eyes may come in different shapes and shades of blue. However, it is important to note that all purebreds have blue eyes. They are also one of the largest domestic breeds of cat. Not only can they reach weights of 9 kg but when you add in their medium to long coat, these gorgeous cats can be the size of a small dog.

Ragdolls have dog-like personalities. They take the best traits from dogs such as loyalty and playfulness. They are fiercely loyal to their families and love being near their humans. Ragdolls will even wait patiently at a door for their owner to come and play. They love to play just like any other cat, they have also been known to play a game of fetch.

Another of the common cat breeds in Australia is the Bengal cat. The Bengal cat is widely known for its sleek and soft coat that contains fur patterns that are marbled and spotted. Both patterns are tri-colored, which makes every single cat unique. The Bengal has spots that have a darker perspective and outline, which makes them similar to Jaguar. Bengal cats are available in different colors such as Snow, Silver, Brown, Blue, Chocolate, Cinnamon, and Charcoal. They have white tummies that are similar to wild cats and eyes are gold or green, but in some cases, they can be blue. Bengals can reach up to 5.5 kg in weight.

Another popular breed is the Abyssinian cat, also known as “Aby” It's a medium-sized cat with a lean, muscular body and a short coat. The Abyssinian has a distinctive “ticked” tabby coat, in which individual hairs are banded with different colors. Coat colors include ruddy, red, blue, cinnamon, or fawn. The eyes are green or gold.

Abyssinians are extremely active and playful, they love to climb, tackle toys, and play interactive games with its owners. Friendly with children and other pets. The Abyssinian has a friendly and outgoing personality, so it loves to play and hates being alone. While this medium-sized cat is very social, its adaptable temperament and playful nature allow it to find fun in tackling toys or racing up and down stairs (or scratching posts) all by itself. 

The Abyssinian makes an excellent addition to an active family. It is recommended, however, to provide your Aby with a companion cat—they’re very social and can become bored if left alone for too long. Get yourself an Aby and soon enough it will be seeking attention from its housemates, no matter the species.

The Burmese cat is a medium-sized cat with a round head, round tips of the ears, round eyes, round chin and even round feet. It is a stocky cat and somewhat compact but is very muscular with heavy boning. The coat of the Burmese is short and glossy. Coat colors include sable, champagne, blue, platinum, lilac, fawn, red, cream, chocolate, cinnamon and tortoiseshell.

Younger Burmese are active, curious cats, and adapt easily to changes. However as they get older, some Burmese can become a bit too placid, preferring to watch rather than get involved in activities. At any age, they love to look at the world around them, their favorite place may turn out to be a window where they can observe the world outside.

Burmese cats are very comfortable with other Burmese but they might not necessarily get along as well with other breeds. Burmese are good climbers and jumpers and love to have access to cat trees and perches. The Burmese is a sturdy, stocky cat and nutrition should be watched to prevent obesity, particularly if the cat does not get enough exercise. While adult Burmese are placid cats, they also tend to be very kittenish and love their daily playtime. They love being adored by their parent and love having their stomach rubbed and being petted.

The Persian cat is a medium- to large-sized breed with a round, flat face shaped like a pansy and a profuse, long coat. The coat comes in almost every color and pattern imaginable, including solid shades, silver and golden, smoke and shaded colors, tabby patterns, particolors and bicolors, and pointed colors (Himalayan). The eyes may be copper, blue, green, blue-green, hazel or odd-eyed.

The Persian cat is known for its very sweet, gentle, calm disposition. Though Persian cats are quite friendly, they require gentle handling. They get along great with kind, respectful kids but would rather be stroked and admired than engage in strenuous activities. Persians also enjoy the company of other cats and gentle dogs. Persians love to sprawl out in a favorite spot in the home with good vantage points to keep an eye on the goings-on in the household.

Persians are homebodies. It’s best to keep them indoors to prevent overheating or tangling up their profuse, long coats. Persian cats love their human family but they don’t mind being left alone for short periods of time as long as they are in their own homes where they feel comfortable and safe.

The Russian Blue cat is a playful, intelligent breed with a striking coat that inspired its name. This cat has green eyes and short dense hair, which is uniformly steel blue with an almost silvery cast, making it one of the most distinctive and beautiful cats in the world. The eyes are green. 

Russian Blues enjoy interacting and playing with their owners, but they also have an aloof quality that makes them less suitable for homes with rambunctious children or pets. Mellow dogs and gentle children are much preferred by this independent breed.

The Russian Blue is a medium-sized cat when it reaches maturity at about a year old. It is friendly and enjoys being near familiar people in calm surroundings. If strangers, rowdy dogs, or overly affectionate kids approach, the Russian Blue is likely to retreat for the safety of a hiding spot where it will stay until the environment has returned to normal.

The Siamese cat is an exquisitely beautiful cat, very affectionate and requires dedication from its parent. The Siamese is a long, elegant cat with a slender but muscular body. The coat is short, glossy and lies close to the body. The beauty of the Siamese cat is the look of the slender body, the blue eyes and the contrast between the color on the body and the darker color of the extremities. This contrast is called color restriction, or, more commonly, pointing.

The Siamese cat is not only beautiful but also highly intelligent. These clever cats can even be trained to walk on a lead. 

With the Siamese having a long, muscular body, weight gain can show quickly. The Siamese will usually show a pot belly after indulging in a day of over-eating. The nutrition must be carefully controlled. The long, slim legs are not made to hold a fat body. Siamese are great jumpers and love heights, so perches and cat trees are very much enjoyed.