Need gorgeous, durable graphics printed on fabrics? Dye sublimation (commonly called dye sub) is most likely going to be your best bet. Fabrics stay more flexible with less chance of the graphics being damaged when compared to screen printing or solvent printing.
You know those old shirts you had… the ones with the letters cracking and coming off? That was not dye-sub. If they had dye sublimation printing back then, that old shirt would still look good. [edit: it seems they did have dye-sub back then. You should have bought a nicer shirt.]
Click on the image on this post. That is a banner we just sent to Hong Kong to be part of a stage set for Bank of America.
You can read more about dye sublimation on Wikipedia.