Pros: small, light, affordable, some models have exceptionally long telephoto capability, most do decent video now, not great, but decent, they are fairly affordable, and they do not draw a lot of notice by people on the street, or uniformed security at stadiums and events. "A few" have large-ish sensors AND extremely long telephoto lenses, and can deliver VERY good ultra-telephoto type images. Virtually all of these that have huge,long zoom ranges have built-in image stabilization. Canon, Nikon, Sony,Fuji, and Panasonic are companies that make some very fine compact and bridge cameras. Looking like "a tourist" or "just an average dad" can sometimes lead to better photos in social situations than walking around with a "big, black camera". Shallow depth of field can be difficult or impossible to get with many compacts; but on the plus side, deep, expansive depth of field is best achieved with a compact camera, so the compact can actually take SOME kinds of photos that just cannot be done with say, a medium format rollfilm camera, or a 35mm SLR,or a FF digital camera.
Cons: might not offer raw image capture option. Lens is usually fixed (not always, but on most the lens is fixed to the body permanently). Small sensors mean highest ISO levels and lowest light performance is not as good as with cameras that have bigger sensors and can use "fast" lenses that have "bright" apertures like f/1.4 or f/1.8 or f/2 as their widest aperture. Battery life is almost such that you will need one, or even two spare batteries to make it through a long shoot, whereas the "big" d-slrs have two-day batteries, and the entry- and mid-level d-slrs have "700-1,000-shot" type battery capacity. Many mirrorless cameras have dreadful 300-450 shot batteries, as pathetic as many compacts have. Fast action shooting performance might not be good enough for extreme sports or action, in some or many cases; that kind of depends.