Nikon's introduction of its Z6 and Z7 cameras might make it the most recent company to hop on board the professional mirrorless-camera bandwagon, but the venerable Japanese firm is hardly the first to take a crack at cutting the flipping mirror of single-lens-reflex cameras out of the optical path.
Olympus and Panasonic were some of the pioneers of the digital mirrorless camera with the Four Thirds and Micro Four Thirds systems. Fujifilm's X System and Sony's Alpha family of cameras are some of the most popular mirror-free picture-takers available today. Why is the industry making the mirror persona non grata, though? Let's take a little trip into the history books.
The vast majority of interchangeable-lens cameras made since 1959 have been single-lens reflex designs, or SLRs. SLRs include a mirror between the rear of the lens and the film or sensor to redirect light through a focusing screen and erecting pentaprism. That system meant that you could, well, focus, compose, and even preview the depth of field in your image at the aperture you chose, since you were looking directly through the lens.
That direct, one-to-one view was the major advance of 1959's Nikon F and other, earlier SLRs over the Leica M and other rangefinder cameras without reflex mirrors: the image in an SLR's viewfinder was always representative of what you would see in the final shot with practically any lens, from the widest possible wide-angles to the longest long lenses.
Rangefinders have endured among photo enthusiasts, however, in part because of the optical quality that's possible when you don't have to design lenses that account for a flipping mirror between the rear of the lens and the film or sensor plane. Rangefinders can be smaller, lighter, and quieter thanks to their less-complex designs, as well, but image quality is the primary reason they still have their adherents.
That advantage is especially true for rangefinders' wide-angle lenses, the rear elements of which can be placed as close to the film or sensor as is practically possible. SLR lens designers have to artificially increase the distance between the rear of the lens and the film or sensor to make room for the mirror and its surrounding infrastructure.
As the article by Bruce Sirovich above notes, that design principle requires complex retrofocus lenses with many glass elements, and complex lens designs tend to be harder to keep sharp, all else being equal. Leica cameras and lenses—the most prominent rangefinder system still in use—have a deserved reputation for being among the sharpest in the world in part because of the inherent advantages the rangefinder camera design offers optical engineers.
Rangefinders have all sorts of inconveniences for practical photography, though. As I alluded to in the first paragraph, one inconvenience is parallax at close distances. By design, rangefinders make the photographer peer through a separate viewfinder with its own perspective, not directly through the lens, so the mismatch between your eye's perspective and the lens' perspective grows as you focus on closer subjects.
This mismatch makes macro (or extreme-close-up) photography practically impossible with rangefinders, barring the use of specialized focusing devices (e.g. Leica's Visoflex). Rangefinders can also show more of the image than you'll actually get in their viewfinders, so it can be difficult to achieve precise compositions at the edges of your frame. Rangefinder lenses can even intrude into the view offered through the finder, marking yet another way it can be more difficult to achieve the precise composition you want.
On top of those issues, rangefinders have a limited set of frame lines in their viewfinders that they can show to define the edges of your composition, so the handy zoom lenses common from SLR makers are practically unheard of in rangefinder systems. Rangefinders also can't show the full view from wide-angle lenses, so you have to compose your wide-angle shots through another, auxiliary finder that's different for every wide-angle lens you have in your bag before focusing using the limited view through the rangefinder window. That separate finder is in yet another position versus the lens itself, so precise composition remains difficult.
Finally, with long lenses (above 85 mm or so), rangefinder photographers have to cope with calibration issues between the focusing system and the lens itself (usually controlled by a mechanical coupling between lens and rangefinder). Rangefinders and lenses that are poorly matched make it hard to get sharp shots, and fixing that mismatch usually requires the purchase of a different lens or sending the camera and lens back to the manufacturer to get them in closer agreement. That's one reason why you don't generally see lenses longer than 90 mm in common use with today's Leicas.
In short, even though rangefinder cameras don't need flippy mirrors or pentaprisms to assist with composing or focusing, the fact that rangefinders can't see through the lens attached to the camera makes them specialized tools, not the Swiss Army knives of photography. That mantle has, for many years, been claimed by the SLR.