Maybe there are more obvious choices.
Sony are innovating and pushing the technology further with their A7 series and sensor technology.
Canon and Nikon have a far greater range of options, and some great features in their high-end cameras.
Olympus and Panasonic have excellent image stabilization technology that can make a notable difference in low light.
Fujifilm don't have the highest resolutions, or the largest sensors, or the biggest following. So why use them?
To get the Canon/Nikon argument out of the way first, I primarily want a smaller camera. I also believe that mirrorless technology is the way forward. Now that electronic viewfinders are so fast and accurate, there is little or no need for an optical system (although admittedly there are still advantages for tracking and shooting fast moving objects). And an electronic viewfinder offers many advantages such as giving a preview of the exposure.
For their size, Fuji cameras feel great to use. They don't have the biggest grip (Sony do a great job here in their A6000 series) but still feel very comfortable in my hand. They're relatively small and easy to carry around for a full day. And the controls just make sense, with fixed physical dials for aperture, shutter speed, exposure compensation and now ISO. Everything easily accessible without going into menus or changing modes.
They look great. I don't mind admitting that it's a factor. I enjoy using products that are well designed aesthetically as well as functionally.
They produce great images. JPEG files are beautiful straight from the camera and they have excellent 'film simulations' to replicate some of their old film styles.
They also provide firmware updates with new functionality to older cameras, which is pretty rare and great to see.
Anything I'd change? If you're listening Fuji, more weatherproofing, touchscreens and smartphone integration please! Thanks.