Assuming you're talking about the optical microscope, the answer is that nobody really knows for certain.
For many years in scientific circles, we were taught it was Anton Van Leeuwenhoek, a Dutchman. Not much credence is given to this theory nowadays despite the fact that most of the public still seem to think he invented the instrument.
A Dutch father and son by the name of Janssen are now usually credited with being the inventors of a type of rudimentary microscope somewhere around 1600, before Van Leeuwenhoek was even born.