The two largest organizations are the Cat Fanciers Association (CFA), which operates mostly in North America, and its European counterpart, Federation Internationale Feline (FIFe).
CFA recognizes 45 cat breeds at present. The newest additions are the Khao Manee and Lykoi. Both allowed to compete in cat shows starting in 2018.
FIFe recognizes 51 cat breeds.
These differing numbers do not mean that FIFe recognizes five more breeds than CFA does. There are quite a few breeds that are unique to each organization; for example, the German Rex is one that FIFe acknowledges, while only the CFA recognizes the Ragamuffin. In general, Europeans seem to be more open to recognizing short- and long-haired versions as different breeds, such as long- and short-haired American curls and Selkirk Rexes, while the CFA says that these are two versions of the same breed.
The total number of distinct breeds recognized by either or both the CFA and FIFe is 64.
Another large body, The International Cat Association (TICA), currently recognizes 73 cat breeds as eligible to compete for the championship title in its shows. Besides, TICA has placed six more breeds on a waiting list of sorts. That is, they are at different stages in the recognition process.
The above gives a rough insight into the number of cat breeds. Many national and regional organizations exist, and the number of breeds they recognize differs, but most acknowledge between 40 and 80. The more intriguing question is, how has this number changed over time?