Every professional observatory built in the last one hundred years uses reflective telescopes.
The last lens-based, refracting telescope used for research astronomy was built before 1900 --- because large lenses become extremely heavy and suffer from a problem called chromatic aberration, which blurs their images. Reflecting telescopes don't suffer from those problems. Because light reflects off the front surface of a primary mirror, the glass can be hollowed out --- lightweighted --- which makes it easier to handle, and since light doesn't travel through the glass, it doesn't suffer from chromatic aberration. This is why all serious astronomical telescopes built in the last hundred years are reflecting telescopes, or reflectors.
Visible
Early astronomers were telescope makers, too; they built instruments to look through.
Visible light is electromagnetic radiation the human eye can detect. It runs from about 400 nm (400 billionths of a meter) to about 750 nm, with some variations from person to person. The first reflectors were built to collect and focus visible light. The telescopes were designed for astronomers to look through and observe. With the advent of, first, cameras and film, and then electronic detectors, the range of these telescopes was extended.
Ultraviolet
Combining information across different wavelength ranges, from ultraviolet to infrared, provides more detailed information.
The ultraviolet (UV) region of the electromagnetic spectrum lies right next to visible light. UV light has shorter wavelengths --- thus higher energy --- than visible light. Although definitions vary, the ultraviolet covers the wavelength range from about 400 nm down to about 100 nm. Some ground-based telescopes have instruments sensitive to ultraviolet radiation, but since the atmosphere absorbs those wavelengths strongly (luckily for the living beings on Earth) most will not even bother looking at wavelengths shorter than 300 nm. The advanced camera for surveys on the Hubble space telescope, though, is sensitive to wavelengths as short as 115 nm.
Infrared
Infrared wavelengths reveal details about the composition of matter that can't be seen in visible light.
On the opposite side of the visible spectrum --- the long-wavelength, lower-energy side --- are the infrared wavelengths. Again, there is some disagreement about what wavelength region is spanned by the infrared, but it covers from about 0.8 µm (a micrometer is one millionth of a meter) out to 100 µm or so. The atmosphere again absorbs a lot of these wavelength, but there are "windows" where the absorption isn't too strong, especially for ground-based telescopes at high altitudes. The coating on the mirrors of most visible light telescopes will reflect at least some infrared light. To access the whole spectrum of infrared light, a telescope needs to be above the Earth's atmosphere, as is the Spitzer space telescope, which is sensitive to wavelengths from 3 µm all the way out to 180 µm.
Extended Regions of the Electromagnetic Spectrum
Radio waves are also electromagnetic radiation, and the big dishes are like mirrors for radio.
Visible light is just a tiny region of the electromagnetic spectrum. Even adding in ultraviolet and infrared wavelengths, there's still a lot more spectrum out there. There are reflecting telescopes that look on the low-energy side at microwave and radio waves and they also look at the high-energy side of X-rays. There are telescopes that look at even higher energy gamma rays, but they aren't reflective telescopes because gamma rays go right through mirrors. Some people say that the optical region is the same as the visible, while some include ultraviolet and infrared. And some include more of the electromagnetic spectrum in the definition of "optical." If you side with that last group, then optical reflecting telescopes look at radiation all the way from X-rays to radio waves.
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