There is no official or standardized answer to this, but many of the big players have undertaken mostly-independent efforts to extend or replace existing type specifications to include color, and it looks like many of those efforts are because of emoji support. Some of these companies’ specifications, while not accepted as standard, have already been implemented in their products.
The most concise-yet-balanced source I could find was the OpenType page on Wikipedia, which I’ll summarize here:
- Apple added an extension in OS X Lion for colored emoji fonts. The implementation used PNG graphics (raster).
- Google did something similar in FreeType 2.5.
- Microsoft added color support in Windows 8.1 with the Segoe UI Emoji font. Microsoft’s implementation uses vector graphics.
- Mozilla and Adobe have proposed adding full SVG support (including color and animation) to OpenType fonts, supported since Firefox 26.