I've been doing a lot of research on this topic lately. Here is what I've found (note that I use "mirrorless" to refer specifically to mirrorless cameras with interchangable lens):
tl;dr: Mirrorless can give better value, but the technology is still catching up, and poor usability in low-light means they are not a replacement for DSLR's.
#Pros of using a Mirrorless
Shorter Flange Focal Distance (FFD) (the actual distance from the sensor to the lens) means that wide-angle lenses can be made smaller, cheaper, lighter, and sharper than what is physically possible for DSLR's. This applies to other lenses as well, to a lesser extent.
Fewer Moving Parts. In any digital system, the mechanical (moving) parts are always the most prone to failure. Because they have fewer moving parts, mirrorless cameras should last longer on average (assuming the same build-quality for everything else).
Audio-free. DSLR's are pretty noisy (sound-wise) when shooting, due to the mirror moving up and down. The noise can sometimes ruin the shot (eg. scare away the animal you're shooting), or get you in trouble (eg. when capturing the silent-moment for someone who has died).
Cheaper. All other things being equal, a mirrorless camera should be cheaper to produce, simply because it has less precision-parts. Lenses of similar quality should also be cheaper, because they can be made with fewer/simpler lens-elements due to the shorter FFD.
What this means is that, if you have $XYZ amount of cash allotted to spend on a camera or lens, having a mirrorless will allow you to get higher-quality parts for your money (Note: This is only mildly true right now, due to supply-and-demand, but as the technology becomes more popular, prices will lower).
This is important for anyone who doesn't swim in a pool of money every morning.
#Cons of using a Mirrorless
Can't be used in low-light. Because there is no optical viewfinder, if the light is too low, the camera basically can't be used. This is the major issue with mirrorless cameras.
Display will never be as "real-time" as a viewfinder. However, displays have gotten so blazingly fast that there is zero practical lag (1ms or less) on the higher-quality screens. The only time I can imagine needing true real-time is when following a high-speed object, like in some sports or nature photography.
Battery life. Mirrorless cameras usually have shorter battery life, due to having to keep the LCD and CCD on all the time. As LCD's, CCD's, and batteries become more efficient, this becomes less of a problem.
However, that is not the complete story. There are some issues I did not include above, either because (A) They are mainly psychological in nature, (B) They have to do with the way the technology is currently implemented, rather than the technology itself, or (C) Are easily worked around.
#Somewhat-Pros of using a Mirrorless
Less Vibration. Even when using a tripod, when the mirror goes up on a DSLR, it causes a small amount of vibration, which can affect the sharpness of the image. However, many DSLR's come with a "mirror up" mode, which prevents this by keeping the mirror up at all times while taking photos and completely negates this issue. The viewfinder cannot be used while the mirror is up, so the DSLR essentially becomes a mirrorless with a longer FFD.
Faster images. Some mirrorless cameras can capture at 60 FPS or more. However, there's no technical reason a DSLR couldn't be built to do the same in mirror-up mode; they just currently don't.
#Somewhat-Cons of using a Mirrorless
No Viewfinder because the mirror is what makes the viewfinder possible to begin with. Some models have a "fake" viewfinder (with an LCD display inside), but these offer no advantage over the larger LCD screen on the back of the camera. I consider this a mainly psychological issue, since the LCD display completely replaces the viewfinder (other than the issues mentioned above). Mostly, people want the viewfinder because it's what they're used to, and because without it...
It looks less professional. This is due to the perception that pros always use the viewfinder, and will certainly change as this technology becomes more popular.
Smaller bodies. The lack of a mirror allows mirrorless cameras to be much smaller and skinnier than DSLR's. However, just because the body can be made smaller does not mean it has to be, and once camera manufacturers realize that not everyone wants a tiny camera, they'll likely start building mirrorless cameras with larger bodies.
Can't use existing DSLR lenses due to the longer FFD expected by these lenses. You can buy an adapter, but then you lose the possible sharpness-bonus.
Might be harder for some people to stabilize while taking hand-held shots because they are used to using a viewfinder rather than holding the camera in front of them. They can improve this simply by practicing: from what I've read, both stances have about equal stability when done correctly.
Current high-grade mirrorless cameras are not on par with high-grade DSLR's. This will of course change as time goes by.
Now, opinion time: Is DSLR a dying breed? I personally don't think so.
Right now, mirrorless cameras have a perception of being non-professional, so Pros don't want them. However, without Pros willing to buy them, the camera manufacturers are hesitant to produce super-high-quality mirrorless cameras, because no one will buy them. It's a sort of bootstrap problem; but as the technology slowly gains acceptance, the issue will solve itself.
Even once the technology catches on, though, mirrorless cameras will supplement, rather than replace, their DSLR's, since each have their own advantages and disadvantages (above). So, what you will probably see 10-20 years from now is Pros with both mirrorless and DSLR cameras in their repertoire, for different occasions.