As always, I'll steer you toward reviews and comparisons. I use DSLRs except when I need something small and compact. Then I use my Sony A6000.

Read comparisons and specs, and decide what features are important to you.
(Reviews) https://www.youtube.com/user/TheCameraStoreTV/videos
http://cameras.reviewed.com/
http://camerasize.com/
http://www.imaging-resource.com/IMCOMP/COMPS01.HTM
http://snapsort.com/compare
http://www.dpreview.com/products/compare/cameras?utm_campaign=internal-link&utm_source=mainmenu&utm_medium=text&ref=mainmenu

After an exhaustive search that took me four years, I bought a Panasonic LUMIX GH4 about 18 months ago.

There are four outstanding mirrorless providers: Fujifilm, Olympus, Panasonic, and Sony. Canon and Nikon are generally behind the others, for various reasons.

Fujifilm uses an APS-C size X-Trans sensor (1.5x crop factor). They have around 20 mostly phenomenal lenses. Their claim to fame is excellent straight-out-of-camera JPEG files that simulate Fujifilm films. Their latest models are excellent, with fine performance, image quality, and very easy to use menus and bodies. They have both rangefinder style and SLR style bodies. MetaBones SpeedBoosters for Fuji are adapters that reduce focal length and improve performance, while adding a full stop or more of effective maximum aperture. So those with lots of dSLR lenses, particularly Canon EF, should look at them.

Olympus and Panasonic share the Micro Four Thirds (Micro 4/3, m43, M4/3) format (2x crop factor). They share the same lens mount, with over 90 native lenses available, and hundreds more that can be adapted (with varying degrees of automation). The native lenses can be 2/3 to 1/4 the weight and bulk of full frame lenses. THAT is what makes the format so appealing. You can stuff a LOT of gear in a small space, which makes travel and nature photography much easier. Lenses are available in a wide array of features and performance, with fast, weather-sealed pro lenses to light, compact travel lenses that work best on the smaller bodies. Voigtlander, Samyang, and Tamron make lenses for M4/3, too. Canon EF lenses can be adapted to Micro 4/3 cameras and retain IS, AF, and auto-aperture control. MetaBones SpeedBoosters for M4/3 are adapters that reduce focal length and improve performance while adding a full stop or more of effective maximum aperture. They are particularly useful for videographers and cinematographers using GH series cameras.

Olympus is known for excellent JPEGs, retro styling, great performance, and in a few models, a mode that makes super high-resolution files from multiple exposures (great for landscapes, products, copying artwork, and other still-life work). They are highly customizable, so the menus can be a bear to navigate. They feature the best IBIS (in-body image stabilization) on the planet, and it works with all lenses. They make a few lenses that have IS in them, and their latest model can use IBIS and lens IS at the same time. They have both rangefinder style and SLR style bodies. The OM-D EM-1 Mark II is their flagship rugged, freeze-proof to -10°C, splash-proof, dust-proof, and fast as a rabbit.

Panasonic is known for excellent raw files, modern styling, great performance, and fantastic video performance. I got the GH4 because I put equal emphasis on stills and video in the work I do (documentation and training). The latest models feature Dual IS (in-body and in-lens stabilizers working in tandem). Panasonic lenses and bodies work together to reduce or eliminate chromatic aberrations. Panasonic has partnered with Leica on a number of their lens designs. Those lenses are outstanding. The Panasonic GH series and some Panasonic lenses are splash-proof and dust-proof. The GH5, due in late March/early April, is freeze-proof to -10°C also. Panasonic makes one notable retro design, the GX8 and GX85. Their G85 and GH5 are really advanced. Panasonic's menus are also extremely well laid-out. Their electronic viewfinders and OLED screens are among the best in class. The G85 and GH4 controls are the easiest to use of any camera I've ever had in my hand, and I've used Canons, Nikons, Pentax, Bronica, Mamiya, Yashica, Beattie, Camerz, Calumet, Hasselblad, and more. Panasonic has worked with a lot of pros over the past few years to help design their flagship.

Sony makes both Full Frame and APS-C mirrorless designs. Their full frame cameras are known for low noise, best low light performance, decent video, and a comparatively few, heavy, and expensive (but excellent!) lenses. Their APS-C cameras are light, excellent performers, too, although the lens weight is still a consideration. The A7 series is full frame, with both high resolution and low light models available. The 6xxx series is APS-C, and the a6500 is their latest. The a6300 is probably the better value. Reviewers tend to score them about the same. Configuring Sony cameras can be a confusing challenge, due to a complicated menu structure. One GREAT feature of the Sony cameras is that most Canon EF series lenses can be adapted to the Sony bodies and retain IS, auto aperture, and slow autofocus (using the correct adapter). MetaBones SpeedBoosters for Sony are adapters that reduce focal length and improve performance, while adding a full stop or more of effective maximum aperture.

Canon's new M5, its third generation mirrorless camera, is getting good, if not great reviews. Canon has been late to the mirrorless game, and has fewer than ten native lenses for their M, M3, and M5 bodies. They do have an adapter for their dSLR lenses, but it has been panned as slow to focus, and inaccurate in its focus. Personally, I'd rather adapt Canon lenses to Fujifilm, Panasonic, or Sony cameras...

Nikon's 1 system has been out for awhile. It is a small, compact, lightweight travel camera that features a CX sensor, about half the size of Micro 4/3. The format has 12 of its own lenses, and an adapter for certain specific Nikkor lenses. I will simply point you elsewhere for commentary on the format, because I've never been a fan. There are some rabid fans out there, however. More power to 'em...

I think you can see that there is something for everyone in the mirrorless camp.

The major complaint about mirrorless from the start was performance. The very latest from Fujifilm, Olympus, Panasonic, and Sony have drastically narrowed the performance gap between high end dSLRs and mirrorless. Electronic viewfinders and processors have improved tremendously in the last few years. That has received plenty of attention. The GH5 will be able to take 6K (18MP) photos at 30 frames per second, at full 4/3 aspect ratio, so you can grab the peak of most any action. It gets faster than that with 4K (8.2MP) photos.

There are many here who pooh-pooh the notion that mirrorless cameras are good for anything. I was once in that crowd — about 2012. No more. My GH4 does what I need done. Most of my work is an even split of video and stills for training documentation, and the vast majority of that is either 1080P video or letter-size documents and PDFs with smaller images in them. Once in a while, I send an image out for a 40x30. I have yet to be disappointed, and I'm picky! Over a decade ago, I set up and ran the digital printing and color correction areas of a large pro lab.

I would encourage anyone interested to go online and watch reviews of all the mirrorless camera lines. Go to http://www.dpreview.com and look up reviews of the various models and their lenses. They arrange their site by brand and date released, so it's easy to view the most recently reviewed models.

Well, the Oly Pen F is the GX8's main competition. Pen F has mediocre autofocus, but looks cooler and has better film simulations for JPEGs. The GX8 has Dual IS, which uses OIS in the Panasonic lenses along with sensor stabilization in the body for up to five stops of shake reduction. The Pen F has IBIS, which provides up to 5.5 stops shake reduction. IBIS is better at shorter focal lengths, while Dual IS is better at tele focal lengths.

The GX8 has better video, but the Pen F doesn't have shutter shock issues. However, the Pen F tends to be unbalanced with larger zoom lenses mounted, and can be a bit of a pain to use with tripods when using fat-barreled lenses. (The GX85 fixed the shutter shock issue by using a new shutter mechanism.) Pen F menus are arguably harder to decipher... Panasonic has a very straightforward layout.

dpreview.com gave both a Silver Award, and a score of 82. It will be interesting to see what Panasonic does with the GX9, and what Oly does with the Pen F II. Both companies have advanced their latest models way beyond the GX8 and Pen F. Micro 4/3 is maturing quite nicely! The good news is, whatever lenses you get for one brand will fit on the other. You might lose a feature or two, but they'll still make images.