How many mainframe computers are in use today?

Mainframes are somewhat less popular now than they were at one time, but they still serve a useful purpose in some applications.

In fact, mainframes have now changed dramatically in size and form, and are not always in the big cabinet-sized boxes we used to associate with the term "mainframe". Indeed, many mainframes have been shrunk down to the size of a large PCI card, and are plugged into a variety of other systems, with special software which allows the mainframe applications to be run on the add-in card, in parallel with the "normal" OS.

Overall, mainframes are slowly becoming rarer, though are in no danger of disappearing anytime soon (possibly in the next 50 years, but don't count on it). A good estimate of the total number of mainframes (include the PCI card and other radically altered mainframe form-factors) is likely in the low 10s of thousands - meaning, probably no more than 20-30,000 worldwide, at best. Compare this to well over 1 billion PCs and PC servers currently in existence, and the 10-20 million other systems (mini-computers, various forms of non-PC microcomputers, etc.).