WTC 7 was the third steel-framed skyscraper that was completely destroyed on 9/11. Unlike the Twin Towers, it was not hit by a plane. Its height was 174 meters, and it had 47 floors. It was located in a block separate from the other WTC buildings, 110 meters from the closest tower. The implosion of WTC 7 is shown below (note the dropping of the penthouse).
Investigations. As this was one of the biggest building disasters in world history, the remains of the skyscraper should have been painstakingly investigated. If the building collapsed in seconds to the ground as a result of fires – as FEMA speculated in 2002 – the significance of the event for building safety, building codes, etc. is enormous. It would have been easy to properly examine the debris from the building, as it landed mostly within its own footprint. This was not done, and the physical research material was quickly removed and destroyed. According to NIST, the governmental agency that is still – well after 6.5 years from the event – trying to come up with a plausible report, “no steel was recovered from WTC 7”. This can be seen as either suspicious or absurd, and I don’t think building disasters are investigated absurdly.
The speed of destruction. WTC 7 fell into a pile of debris in approximately 6.5 seconds. The corresponding free fall time is 5.95 seconds, while an apple dropped from the roof would have taken 7 seconds to fall to the ground (Kurttila 2005; the exact time varies with the air resistance of the object). The 80 steel support pillars of the building, therefore, did not in practice resist the destruction. However, destroying the support structures throughout the floors of the 174-meter building demanded energy that would have been away from pure kinetic energy; in other words, gravitational destruction of those structures would necessarily have slowed down the collapse. No slowing down required by destruction work can have taken place within the short time it took WTC 7 to collapse. To simplify: the roof came down as if mere air (and not 47 stories) had separated it from the ground. This can only be explained by the removal of structural resistance in a controlled demolition. In controlled demolitions, the roofs of highrises typically reach the ground in a time that is slightly longer than free fall. 
The sudden onset and symmetry of the destruction. WTC 7 dropped suddenly straight down. This means that the 80 support pillars had to give in simultaneously. To believe that random fires on separate floors and damage to one side caused the sudden vertical collapse is to believe in a miracle (as pointed out by professor David Ray Griffin). Moreover, achieving the outcome of a controlled demolition by means of matches and damaging one side of a highrise would mean that companies specialized in controlled demolition would have to start thinking about new business ideas.
Hot debris. According to NASA’s thermal imaging, the surface temperature of the WTC 7 debris pile exceeded 700 degrees Celcius – five days after the destruction. Residual temperatures like this cannot be explained by fires or gravitational collapse. The latter can only result in a few degrees' increase in temperature.
Molten and vaporized steel. As in the case of the Twin Towers, molten steel was reportedly found in the remains of WTC 7. Some steel samples that the researchers did manage to examine were also partially vaporized. In a New York Times interview, professor Jonathan Barnett points out that fires cannot explain this. Evidently, not even these samples were kept, and NIST has ignored this finding. Molten steel can be explained by explosives but not by fires, as their temperature simply cannot rise anywhere close to the melting point of steel, let alone the temperature required by vaporization.
Expert statements. The Dutch demolition expert Danny Jowenko, who owns a demolition firm and has been in the business for 30 years, regards it as certain that WTC 7 was demolished. His view is shared by numerous architects, engineers and other demolition professionals – see Architects and Engineers for 9/11 Truth, http://www.ae911truth.org/, and this page exemplifying people with demolition expertise who question the official account.
The testimony of first responders. Several rescue personnel have testified to being told that WTC 7 would be brought down.
 Another way of looking at this:
1) In a vacuum, an object falls the height of WTC 7 in 5.95 seconds. An object falling in a vacuum does not crush, twist or displace anything – not even air. If WTC 7 had collapsed in 5.95 seconds, not even air (let alone the rest of the building) would have separated its roof from the ground.
2) WTC 7 collapsed into a rubble pile in approximately 6.5 seconds.
3) As an object falling freely does not crush or twist anything, the time available for gravitational crushing or twisting of the building's ~80 steel columns throughout its 47 floors is slightly over 0.5 seconds (6.5 - 5.95 seconds) – or, if you like, ~0.01 seconds per floor ( 0.5 / 47).
4) The 0.5 seconds is not enough even in theory for a total gravitational collapse of a 174-meter highrise. Ergo, the structural resistance was removed by means of explosives.