The question of who invented the microscope is a matter of historical debate. There isn’t 100% certainty on the microscope invention, because it was not that well documented during its time, but we know for certainty that it occured around the turn of the 17th century.
Furhermore, we are certain that that the first type of microscope invented was the most common one: the optical microscope (which contains a lens that greatly magnifies objects).
A few names regularly crop up of potential inventors who invented the microscope. There is a Dutch spectacle maker named Hans Janssen who, together with his son, was said to have originated the concept of the microscope back in 1590. Another name is Hans Lippershey who is credited with inventing the telescope (which is designed to make objects in the far distance much easier to see), and some people suggest he had a role to play in microscopics as well. But the link is not that strong.
Perhaps the most famous person who invented the microscope, or should we say holds the claim to the invention, is none other than the Italian scientist Galileo Galilei. Galileo is said to be a major force behind the scientific revolution, and is sometimes referred to as the father of observational science and astronomy. In 1609 Galileo invented what he termed the ‘little eye’, which was an apparatus that made use of convex and concave lenses to view objects unseeable by the human eye. A friend of Galileo’s, a German named Giovanni Faber, termed this the ‘compound microscope’ and the name ‘microscope’ has stuck since then.
But it did not simply end then, as it took a while for the microscope to become part of scientific enquiry, research and culture. There were many more people who invented improvements to the microscope, amongst them Cornelius Drebbel, who used double lenses that were concave, Anton van Leeuwenhoek attunes the apparatus for use on biological organisms, and Ernst Abbe in the mid 1860s drastically improved the design of the microscope. Even today there are still improvements and modifications of the observational apparatus.