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Black holes are a finicky lot. They are thought to exist at the center of most active galaxies and are predicted to exist by general relativity. What we theorize black holes to be are supermassive objects "power[ing] galactic nuclei by converting the gravitational energy of accreting matter into radiation" (http://www.nature.com/nature/journal/v455/n7209/full/nature07245.html). But, there's a problem currently with proving the existence of black holes - we cannot see them. We have seen the effects of intense gravity (See the article about Stellar Motion around the black hole located at Sagittarius A* at our galaxy's center.), but nothing definitive yet as to what exactly is the cause. Theoretically it is possible to observe the presence of a black hole by a process known as "gravitational lensing," an effect predicted by general relativity, whereby light from some source travels from the source and is bent by the extreme gravity of an object, such as a black hole or a galaxy. In the case of a black hole, as the black hole comes between some distant bright object and an observer the image of the object will grow in size, making it appear larger, and brighter due to the change in angular size while in line.
However, there is a tremendous amount of electromagnetic radiation emitted from the activity near black holes, and so the most promising possibilities in observing black holes are in the range of x-ray and radio frequencies. What we are actually trying to observe is a relatively small silhouette amongst massive amounts of bright, hot matter. Quasars, which are massive conglomerations of bright, hot matter located at the center of some very distant galaxies, are currently thought to be caused by the gravity of black holes at the center, feeding off an accretion disk of matter and subsequently converting that energy to radiation. Inconveniently for us, this incredible amount of matter near a black hole also serves the purpose of clouding most any imaging of the black hole silhouette.
X-ray telescopes and radio telescopes are our best bet currently for observing black holes. The most recent efforts in imaging black holes have been attempts to image Sagittarius A*, the supermassive black hole thought to occupy the center of the Milky Way.
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Efforts in Radio Astronomy
The most promising effort yet in radio astronomy to see the Sagittarius A* is the linking of three radio telescopes over a stretch of 2,800 miles (one in Hawaii, one in Arizona, and one in California), which forms essentially one large radio telescope. This is done using a technique called VLBI, or very long baseline interferometry, which unites three simultaneous observations, creating one image from one enormous "virtual" telescope with incredible sensitivity, capable of imaging details 1,000 times finer than the Hubble telescope.
The telescope is working at frequencies near 1 millimeter, a wavelength that cuts through much of the cloudy matter near Sagittarius A*. There have been images taken already, although the results thus far have been ambiguous as to what actually is at the center of Sagittarius A*, although evidence is hinting that whatever it is it does have an event horizon, and is not glowing with massive amount of infrared radiation, both bits of evidence are what we would expect if there is a black hole there.
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Page 2 of the article about attempts to image black holes concludes the discussion of using radio waves for observation and presents the advantages of using x-rays, using NASA's Chandra X-Ray Observatory among others, to peer into the veil of matter that surrounds a black hole.
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Efforts in Radio Astronomy (continued)
Efforts will continue with VLBI telescoping in the future and astronomers will continue to attempt to sharpen the image at shorter wavelengths, hopefully to near .87 millimeters. Also, being built currently is Atacama Large Millimeter/Submillimeter Array (ALMA) in Chile which has a total of 66 antennae, operating at millimeter and submillimeter wavelengths--between .3 and 9.6 millimeters--blurring what are traditionally known as radio and infrared waves. As well with this observatory astronomers hope to be able to detect the spin of the black hole by watching for "hotspots" being dragged around in the warped space-time near the black hole, an effect known as frame dragging; and this information can also be used to provide information about the history of Sagittarius A*, as what it swallows will determine the rate of spin.
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X-ray telescopes are observing the x-rays emitted from matter as it heats during its activity near a black hole. X-ray observatories have been taking images of supermassive black holes including images of Sagittarius A*, and the observations have been fruitful, although they have not yet produced a definitive image of the black hole silhouette (see below). What the observations of Sagittarius A* have told us is that the black hole we believe is at the center of our galaxy has shown erratic behavior in the form of flaring and dimming. This action is taking place near the event horizon, and accounts for the relatively low amount of radiation emission coming from Sagittarius A*. This black hole is estimated to have 4 millions times the mass of our sun, and yet the radiation emitted is billions of times weaker than the radiation coming from other black holes.
The flaring was observed by an x-ray telescope utilizing a process called light echo where the telescope observes changes caused by x-ray pulses from Sagittarius A* in a cloud nearby, called Sagittarius B2. The process can better be described as follows: "When the x-rays reach the cloud [Sagittarius B2], they collide with iron atoms, kicking out electrons that are close to the atomic nucleus. When electrons from farther out fill in these gaps, the iron atoms emit x-rays. But after the X-ray pulse passes through, the cloud fades to its normal brightness" (http://chandra.harvard.edu/press/08_releases/press_041608.html). However nearby, the x-ray pulses observed took about 300 years to reach Sagittarius B2, so the information recorded from there is 3 centuries old, and the information we receive from Sagittarius B2 takes about 26,000 years to reach us on Earth (that being the approximate distance to the center of our galaxy, with the x-rays traveling at the speed of light).
The Chandra X-ray Observatory is one tool being used prominently in the effort to image black holes. Chandra is a NASA - Smithsonian Astrophysical Observatory satellite in high orbit around Earth, actually orbiting at over one-third the distance from Earth to the Moon. There are also several x-ray telescopes operated by the Japanese Aerospace Exploration Agency (the newest being Suzaku, launched in 2005, older ones being Hinotori, Tenma, Ginga and Asuka) searching for x-ray emissions from astronomical bodies.
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1. Gravitational Lensing: http://en.wikipedia.org/wiki/File:Gravitational_lens-full.jpg
2. Sagittarius A*: www.space.com