The ancient Egyptians did not allow the export of cats. This was partly because they were so useful for a society based on agriculture and partly due to the respect accorded the cat in Egyptian society. Not only was the cat semi-divine, but as such they could not be the property of anyone other than the pharaoh (whose divine status matched or exceeded that of the cat). There is evidence that cats were smuggled out of Egypt, and we know that on at least one occasion the Egyptian army was despatched to bring them back home! Despite this, the cat population soon spread throughout the Mediterranean. Genetic evidence now suggests that the domestic cat was already on the way to conquering Europe and Asia by this time and that any attempt to prevent this from occurring was doomed to failure.
The archaeological record confirms that by 1700 BC cats regularly appeared in domestic poses in Israeli art and by 1400 BC the domestic cat was firmly established in Greece (although never as popular there as in either Egypt or Rome). By at least 1000 BC cats had travelled northwards across the Mediterranean aboard ships (most likely with Phoenician traders) and by 500 BC the domestic cat was firmly established in southern Europe. Traders also carried the cat to China and Japan.
Alexander the Great conquered Egypt in 332 BC and brought the domestic cat to Greece. Before this time, the ferret (or polecat) was their favourite pet and mouser! The Romans treated cats as exotic pets as well as working animals, although the mongoose was their preferred rat-catcher. During expansion of the Roman Empire they imported numerous domestic cats into Britain (cat footprints have been discovered in clay tiles in Chelmsford, then known by the Roman name of Caesaromagus). However, there is some evidence that the Phoenicians had already brought the cat to Britain whilst trading in tin with the Cornish peoples and the Vikings are also credited with carrying cats to Britain.
By 1000 AD rats and other vermin were becoming increasingly problematic for rapidly growing cities and there was a huge demand for cats to help control their numbers. The domesticated cat was also well established in Scotland by this time - having mounted an invasion rather more successful than that of the Romans! Cats were also popular with the catholic church at this point and often acted as guardians of the store in monasteries.
There is archaeological evidence that the Vikings "discovered" and briefly settled the eastern coast of North America at around 1000 AD and it is often proposed that they took the domesticated cat with them. One much later story (from "Natural History" by Count de Buffon published in 1767) claimed that there were native wild cats already living in America when it was "discovered" by Europeans some time later and that Christopher Columbus was given one such wild cat by a hunter. However, the term "cat" was applied to cat-like non-felid mammals such as the pine marten and the bobcat, and the story specifically states that the animal was wild and not domesticated. In fact, it is often suggested that the animal was actually a racoon! So far, domestic cat bones have not been discovered which pre-date the arrival of European settlers, so it seems reasonable to assume that it was the Europeans who brought the domestic cat to the Americas.
Although there are two native species of cat in South America who can be tamed, (Geoffroys Cat and the Margay) they are both forest cats who did not come into regular contact with a static agrarian civilisation (as they did at a very early stage in Egypt, the Middle East and Asia) and so they never became domesticated. The cats who did live in the areas where Central and South American peoples built their cities may not have been so amenable to domestication. In any case, there is no evidence of the domestication before the advent of the Europeans.
During the European colonisation of America, thousands of cats earned their passage by killing vermin on ships. Polydactyl cats (cats with more than five toes on each front paw and/or more than four toes on each back paw) were very popular as they were thought to be lucky mascots and so there is a higher than average proportion of polydactyl cats along the eastern coast of the USA where the colonists first settled.