History of the microscope
A 17th century compound microscope, from an engraving in Robert Hooke's Micrographia.
It is impossible to say who invented the compound microscope. Dutch spectacle-makers Hans Janssen and his son Zacharias Janssen are often said to have invented the first compound microscope in 1590, but this was a declaration by Zacharias Janssen himself halfway through the 17th century.
The date is certainly not likely, as it has been shown that Zacharias Janssen actually was born around 1590. Another favorite for the title of 'inventor of the microscope' was Galileo Galilei. He developed an occhiolino or compound microscope with a convex and a concave lens in 1609. Galilei´s microscope was celebrated in the ´Lynx academy´ founded by Federico Cesi in 1603. Francesco Stelluti´s drawing of three bees were part of pope Urban VIII´s seal, and count as the first microscopic figure published (see Stephen Jay Gould, The Lying stones of Marrakech, 2000). Christiaan Huygens, another Dutchman, developed a simple 2-lens ocular system in the late 1600's that was achromatically corrected and therefore a huge step forward in microscope development. The Huygens ocular is still being produced to this day, but suffers from a small field size, and the eye relief is uncomfortably close compared to modern widefield oculars.
Anton van Leeuwenhoek (1632-1723) is generally credited with bringing the microscope to the attention of biologists, even though simple magnifying lenses were already being produced in the 1500's, and the magnifying principle of water-filled glass bowls had been described by the Romans (Seneca). Van Leeuwenhoek's home-made microscopes were actually very small simple instruments with a single very strong lens. They were awkard in use but enabled van Leeuwenhoek to see highly detailed images, mainly because a single lens does not suffer the lens faults that are doubled or even multiplied when using several lenses in combination as in a compound microscope. It actually took about 150 years of optical development before the compound microscope was able to provide the same quality image as van Leeuwenhoek's simple microscopes. So although he was certainly a great microscopist, van Leeuwenhoek is, contrary to widespread claims, certainly not the inventor of the microscope.