How many cat breeds are currently recognized?

It depends on the cat association. By "recognized" you mean recognized by a cat association. There are many cat associations. They have their own ways of working. They accept and recognize different cat breeds and even categorize cat breeds differently. Although there is a central core of cat breeds that are universally recognized.

In answer therefore to the question, "How many cat breeds are currently recognized?" you have to check the cat associations. Cat associations also categorize cat breeds into different levels. The largest, the Cat Fanciers' Association (CFA) recognize 43 cat breeds as far as I can tell. The International Cat Association (TICA), at championship level, recognize 56 cat breeds .  The other categories at TICA are as follows:

• Non Championship Breeds - 2
• Advanced New Breeds - 4
• Preliminary New Breeds - 8

These figures are correct at the date of this post (Nov 2011). The situation changes slightly over time mainly because new breeds are developed or sometimes the categorization is amended.

The premier cat association in the UK - GCCF - have a different method of categorising cat breeds. Overall there are over 100 cat breeds as some are on the fringes and rare - 104 are shown in the video below but not all are recognised. There are many more dog breeds. See a cat breed quiz.