History books generally identify the inventor of the telescope as one Hans Lippershey, an eyeglass maker in the Netherlands in the late 16th century. The story goes that Lippershey saw a couple of children playing with lenses in his shop, and overheard them exclaim that looking through the lenses made a nearby building seem larger.
Lippershey experimented a bit further, and built a device he called a "looker," using a convex objective lens and a concave eyepiece. Galileo snagged an early version of the telescope as it spread through Western Europe, and improved the design to make the first observations of the moons of Jupiter, among other momentous findings.
But Johannes Kepler suggested the instrument could be improved even more in 1611 by using a convex eyepiece, resulting in a wider field of view. Nor was it necessary any longer to plate the eyepiece so close to the eye of the observer. The only disadvantage: the resulting image is inverted. Astronomers adapted accordingly. The first Keplerian telescopes were believed to appear around 1631.
Of course, there is evidence that Lippershey may not have been the first to build a telescope after all. His 1608 patent application was denied ...