How many types of telescopes are there in the world?

That's a pretty hard question to answer. Someone already pointed out the three primary optical designs: refractors, reflectors, and catadiotropics. Within those classes are a multitude of different designs. read more

The following are lists of devices categorized as types of telescopes or devices associated with telescopes. They are broken into major classifications with many variations due to professional, amateur, and commercial sub-types. Telescopes can be classified by optical design or mechanical design/construction. read more

About 100 of them are non-optical telescopes, which gets you right into the 400 ballpark for the number of optical research telescopes around the world. Keep in mind, that there's no solid dividing line between a named telescope at a well-known observatory, and a hand-held device used by a backyard astronomer. read more