With the recent announcement of iOS 11, Apple has delivered on their WWDC promise of new file formats for stills and video. Whether they become industry wide standards is too early to tell, suffice to say that there is finally something potentially better than JPEG and MPEG-4.
HEIF which despite the rules of proper grammar rhymes with BEEF stands for High Efficiency Image Format, a name which unlike JPEG may actually tell you something about itself. A HEIF file of an image shot on a camera will be physically smaller than a JPEG image shot on the same camera. This space saving is less about the the cost of storage and more about the larger files resulting from the computational photography work being done at time of image capture. Episode 75 of Make Better Photos and Videos discusses computational photography.. Where JPEG supports only 8 bit images, HEIF can support 10 bit images and that means richer colour depth because of a significant increase in bit depth. In JPEG there is a maximum colour palette of 16.7M colours. In HEIF, the ceiling is just over 1 billion colours.
To achieve the smaller size, compression is used. This word brings out the shakes and twitches in serious creatives. Without quantitative data based on a large volume visual analysis, the time since the development of the JPEG compression algorithms and the significant enhancement in compute capabilities, the absolute worst case is that 10 bit HEIF will still be an improvement over 8 bit JPEG. I don't know this for sure but if HEIF gets legs outside of Apple we will all know soon enough. HEIF is not Apple designed, but came from the Motion Picture Experts Group.
Another tangible benefit of HEIF is that it can store multiple images such as brackets, or focus stacks, along with much richer metadata in a single file. This will aid image management a great deal.
If you've switched to iOS 11, HEIF is the default still image format, although you can switch to JPEG if you'd like. This will only matter to Windows 10 users, because at the moment Windows does not have a HEIF format reader. macOS High Sierra (just released) reads and writes HEIF natively. For everything else the Mac will convert HEIF to JPEG transparently.
The bigger question is whether other camera companies will adopt the HEIF format. Time will tell on that one, and it might take time, given how glacial traditional camera vendors have been when it comes to innovation.
HEVC is video equivalent to HEIF. High Efficiency Video Codec is the evolution of the MPEG-4 codec offering again, smaller files and a richer colour capacity. iOS11 and macOS High Sierra natively support HEVC, conversion to MPEG-4 is required for use on other platforms. The value proposition here is more about space saving than resolution improvement because the driving factor is increasing ubiquity of 4K UHD video. These files are much larger than Full HD and if computational process are used in the capture of the video, that further increases the file size. There is still the improvement in colour bit depth, but until we see HEVC out of a large sensor video camera, it will be near impossible to determine if we can actually see the physical improvement in colour depth.
HEIF and HEVC are new formats, not minor enhancements to existing ones. As yet neither Microsoft nor Android support them natively, although some research shows that Google Photos already supports HEIF directly.