A mirrorless camera is one that is most similar in function to a digital SLR in that it can accept different types of lenses and house manual control dials for changing exposure settings.
However, unlike a digital SLR (DSLR), mirrorless cameras don't have a mirror box -- hence the name 'mirrorless' -- which is used in a DSLR to bounce a scene from the lens up into an optical viewfinder so that you can frame your shot. The mirror in a DSLR sits in front of the sensor.
In a mirrorless camera, there is no mirror. The lens sits closer to the sensor, and you frame your shot by looking either through the LCD screen on the back of the camera, or through an attached electronic viewfinder (instead of an optical one).
The biggest advantage of mirrorless cameras is that, because they don't have a mirror box, they can be a lot smaller than a digital SLR -- some models are only just a little bigger than a typical compact camera. This makes them a much more attractive proposition for anyone who would like to carry an advanced camera with similar manual controls to a DSLR camera, but without the inevitable bulk that is present in a DSLR. However, you will still have to carry around a couple of lenses, or at least one versatile zoom lens, just like you would with a digital SLR.
Manufacturers currently offering mirrorless cameras are Sony, Olympus, Panasonic, Fujifilm, Samsung, Canon, and Nikon. Prices can vary greatly for these types of cameras, with professional-calibre models often found for close to $2000. Furthermore, the lenses for such cameras can be expensive in general. The best value for money can be had from twin lens kits, which come with a combination of wide angle and telezoom lenses that can allow you to cover most bases when it comes to common photography tasks, from sweeping landscapes, to closer-up details.
As with all camera ecosystems, you will need to find one that has the lenses that suit your style of photography, and then figure out how much you can expect to pay for the available lenses. Just because mirrorless cameras are smaller, it doesn't mean the prices are always smaller.
Perhaps the best part about mirrorless cameras is that they have been created with mainstream users in mind. Many come with highly capable automatic modes, lots of scene modes, sometimes 'art' modes, and they usually have clean control layouts that make them simple to operate. On many models, there is a 'what you see is what you get' live view of the scene you are about to capture, which can make using the camera a lot more intuitive when it comes to changing exposure settings.
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