When you save your model, any paint layers are saved in the same directory as your .mud file in a subdirectory with the same name as your .mud file with a -files extension (for example, <filename>-files).
When you save your model using the .fbx file format, any paint layers are save in the same directory as your .fbx file in a subdirectory with the same name as your .fbx file with a .fbx-files extension (for example, <filename>.fbx-files).
By default, one image is saved for each paint layer which contains your paint work unless multiple UV tiles exist. In that case, one image is stored per UV tile. In addition, a background image that represents the background color for each channel (diffuse, specular, gloss) is saved to the files subdirectory.
If you want to save one individual paint layer, see Export a paint layer.
If you want to export paint layers to perform further work using Adobe Photoshop, see Create and edit paint layers using Photoshop for features that provide a custom workflow between the two software applications.